StudySmarter - The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

Limits

- Calculus
- Absolute Maxima and Minima
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Disk Method
- Divergence Test
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Improper Integrals
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Differential Equation
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- One-Sided Limits
- Optimization Problems
- P Series
- Particular Solutions to Differential Equations
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Tangent Lines
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal´s Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Comparing Data
- Conditional Probability
- Correlation Math
- Cumulative Frequency
- Data Interpretation
- Discrete Random Variable
- Distributions
- Events (Probability)
- Frequency Polygons
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Testing
- Large Data Set
- Linear Interpolation
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Random Variables
- Sampling
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Measures
- Tree Diagram
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

Have you ever heard the saying “*close only counts in horseshoes and hand grenades*”? Well, it turns out, this isn't entirely true. Close, or **nearly reaching** a target, also counts in calculus – when dealing with **limits**, that is!

The basic concept of a limit in mathematics is essential to your understanding of calculus.

Limits are all about determining how a function behaves as it approaches a specific point or value.

This concept has been around for thousands of years; early mathematicians used this concept to find better and better **approximations **of the **area of a circle**, for example.

The formal definition of a limit, however, has only been around since the 19^{th} century. So, to begin your journey to understand limits, you should start with an intuitive definition.

To find an intuitive definition of a limit, you must first have a function (or several functions) about which you wish to know more details.

Take a look at the graphs of the following functions:

\[ f(x) = \frac{x^{2}-4}{x-2}, \; g(x) = \frac{|x-2|}{x-2}, \; \mbox{ and } \; h(x) = \frac{1}{(x-2)^{2}} \]

You want to pay attention to the behavior of these graphs at and approaching the value of \( x=2 \).

The graphs of these functions show their behavior at and around \( x=2 \). After observing them, can you see what they have in common?

They are all undefined when \( x=2 \)!

- But if that is all you say about them, you don't get very much information, do you? If you are given only this information, then, for all you know, all three of these functions could look identical. Based on their graphs, however, you know this isn't the case.

So, how can you express the behavior of these graphs more completely?

- With the use of
**limits**, of course!

Now, take a closer look at how \( f(x) = \frac{x^2-4}{x-2} \) behaves near \( x = 2 \). Notice that as the values of \( x \) approach \( 2 \) from either side of \( 2 \), the values of \( f(x) \) approach \( 4 \).

To state this fact in mathematical terms, you would say: “the limit of \( f(x) \) as \( x \) approaches \( 2 \) is \( 4 \)”.

This statement is represented in mathematical notation as:

\[ \lim_{x \to 2} f(x) = 4. \]

From here, you can start to develop your **intuitive definition of a limit **– by thinking of the limit of a function at a number \( a \) as being the real number \( L \) that the functional values approach as its \( x \)-values approach \( a \), provided that the number \( L \) exists. More formally, this can be written as:

Let \( f(x) \) be a function that is defined at all values in an open interval containing \( a \) (possibly except \( a \)), and let \( L \) be a real number. If **all **values of \( f(x) \) approach the real number \( L \) as the values of \( x \) – except \( x = a \) – approach the number \( a \), then you can say that **the limit of \( f(x) \) as \( x \) approaches \( a \) is \( L \)**.

Or, more simply:

As \( x \) gets closer and closer to \( a \), \( f(x) \) gets closer and closer and stays close to \( L \).

The idea of the limit is represented using mathematical notation as:

\[ \lim_{x \to a} f(x) = L \]

As you can see, just getting close to – or approaching – a point is how limits work! To develop and understand the key aspects of calculus, you first need to be comfortable with limits and the fact that approximations – or getting close to, or approaching, the desired value – are the basis of calculus. So, now you can change the saying from:

- “
*close only counts in horseshoes and hand grenades*” to - “
*close only counts in horseshoes, hand grenades*, and calculus”!

Before diving into algebraic methods, the next step to take intuitively is to develop a way for solving limits by **estimating** them. You can do this in one of two ways:

Solving a limit using a table of functional values

Solving a limit using a graph

To solve a limit using a table of functional values, you can use this problem-solving strategy.

__Strategy – Solving a Limit Using a Table of Functional Values__

- If you want to solve the limit: \( \lim_{x \to a} f(x) \), you start by making a table of functional values.
- You should choose \( 2 \) sets of \( x \)-values – one set of values approaching \( a \) that are less than \( a \), and one set of values approaching \( a \) that are greater than \( a \). The table below gives an example of what your table could look like.
Values Approaching \( a \) that are \( < a \) Values Approaching \( a \) that are \( > a \) \( \bf{ x } \) \( \bf{ f(x) } \) \( \bf{ x } \) \( \bf{ f(x) } \) \( a - 0.1 \) \( f(a - 0.1) \) \( a + 0.1 \) \( f(a + 0.1) \) \( a - 0.01 \) \( f(a - 0.01) \) \( a + 0.01 \) \( f(a + 0.01) \) \( a - 0.001 \) \( f(a - 0.001) \) \( a + 0.001 \) \( f(a + 0.001) \) \( a - 0.0001 \) \( f(a - 0.0001) \) \( a + 0.0001 \) \( f(a + 0.0001) \) Add more values if you need to. Add more values if you need to.

- You should choose \( 2 \) sets of \( x \)-values – one set of values approaching \( a \) that are less than \( a \), and one set of values approaching \( a \) that are greater than \( a \). The table below gives an example of what your table could look like.
- Next, look at the values in each of the columns labeled \( f(x) \).
- Determine if the values are approaching a single value as you move down each column.

- If both columns approach a common value, then you can say that\[ \lim_{x \to a} f(x) = L. \]

You can extend the problem-solving strategy above to solve a limit using a graph.

__Strategy – Solving a Limit Using a Graph__

- After following the above strategy, you can confirm your result by graphing the function.
- Using a graphing calculator (or other software), graph the function in question.
- Make sure the functional values of \( f(x) \) for the \( x \)-values near \( a \) are in the graphing window.

- Move along the graph of the function and check the \( y \)-values as their corresponding \( x \)-values approach \( a \).
- If the \( y \)-values approach \( L \) as the \( x \)-values approach \( a \) from both directions, then\[ \lim_{x \to a} f(x) = L. \]

For more details and examples, please refer to the articles on finding limits and finding limits using a graph or table.

While the two techniques above are intuitive, they are inefficient and rely on too much guesswork to get the job done. But how can you progress past these methods?

Well, you will need to learn methods to solve, or evaluate, limits that are more algebraic in nature.

And how can you do that? First, you must know about two special limits; they provide the foundation of the algebraic methods to solve limits.

Ah, but what is so special about these two limits? These two limits are also known as **basic limits**, as they provide the basis for the limit laws. When you look at the graphs below, what do you notice?

Based on these graphs, you can write out, algebraically, what the limit of these functions are. The algebraic interpretations of these are summarized in the theorem below.

Let \( a \) be a real number. Let \( c \) be a constant. Then:

\[ \begin{align}1. \; & \lim_{x \to a} x = a \\2. \; & \lim_{x \to a} c = c\end{align} \]

You can observe the following about these two limits:

- Notice that as \( x \) approaches \( a \), so does \( f(x) \).
This is because \( f(x) = x \).

Therefore, \( \lim_{x \to a} x = a \)

- Consider the table:
Values Approaching \( a \) that are \( < a \) Values Approaching \( a \) that are \( > a \) \( \bf{ x } \) \( \bf{ f(x) = c } \) \( \bf{ x } \) \( \bf{ f(x) = c } \) \( a - 0.1 \) \( c \) \( a + 0.1 \) \( c \) \( a - 0.01 \) \( c \) \( a + 0.01 \) \( c \) \( a - 0.001 \) \( c \) \( a + 0.001 \) \( c \) \( a - 0.0001 \) \( c \) \( a + 0.0001 \) \( c \) - Notice that for all values of \( x \) – whether they are approaching \( a \) or not – the values of \( f(x) \) remain constant at \( c \).
- Therefore, \( \lim_{x \to a} c = c \)

Building on these first two basic limit rules, the limit rules (also called limit laws) are listed below.

Let \( f(x) \) and \( g(x) \) be defined for all \( x \neq a \) over an open interval containing \( a \). Assume that \( L \) and \( M \) are real numbers, such that:

\[ \lim_{x \to a} f(x) = L \]

and\[ \lim_{x \to a} g(x) = M \]

Let \( c \) be a constant. Then the following are true:

**Sum law for limits**:

\[ \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = L + M \]

**Difference law for limits**:

\[ \lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) = L - M \]

**Constant multiple law for limits**:

\[ \lim_{x \to a} (c \cdot f(x)) = c \cdot \lim_{x \to a} f(x) = cL \]

**Product law for limits**:

\[ \lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = L \cdot M \]

**Quotient law for limits**:

\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{L}{M} \mbox{ where } M \neq 0\]

**Power law for limits**:

\[ \lim_{x \to a} (f(x))^{n} = \left( \lim_{x \to a} f(x) \right)^{n} = L^{n} \mbox{ for every positive integer } n \]

**Root law for limits**:

\[ \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} = \sqrt[n]{L} \mbox{ for all } L \mbox{ if } n \mbox{ is odd, and for } L \geq 0 \mbox{ if } n \mbox{ is even} \]

Keep in mind that there are other limit laws – the squeeze theorem and the intermediate value theorem. Please refer to those articles for more information.

When you work through the following example, remember that for the limit to exist, the functional values must approach a single real number value; otherwise the limit does not exist.

__Evaluating a Limit that Does Not Exist (DNE)____ Due to Oscillations__

Try to evaluate

\[ \lim_{x \to 0} sin \left( \frac{1}{x} \right) \]

using a table of functional values.

**Solution**:

- Create a table of values.
\(\bf{x}\) \(\bf{sin\left(\frac{1}{x}\right)}\) \(\bf{x}\) \(\bf{sin\left(\frac{1}{x}\right)}\) \(-0.1\) \(0.54402\) \(0.1\) \(-0.54402\) \(-0.01\) \(0.50636\) \(0.01\) \(-0.50636\) \(-0.001\) \(-0.82688\) \(0.001\) \(0.82688\) \(-0.0001\) \(0.30561\) \(0.0001\) \(-0.30561\) \(-0.00001\) \(-0.03575\) \(0.00001\) \(0.03575\) \(-0.000001\) \(0.34999\) \(0.000001\) \(-0.34999\) - Carefully examine the table. What do you notice?
- The \( y \)-values aren't approaching any value. So, it seems like this limit doesn't exist. Before coming to this conclusion, though, you should take a systematic approach.
- Consider the following \( x \)-values for this function that approach \( 0 \):\[ \frac{2}{\pi}, \frac{2}{3\pi}, \frac{2}{5\pi}, \frac{2}{7\pi}, \frac{2}{9\pi}, \frac{2}{11\pi}, \cdots \]
- Their corresponding \( y \)-values are:\[ 1, -1, 1, -1, 1, -1, \cdots\]

- The \( y \)-values aren't approaching any value. So, it seems like this limit doesn't exist. Before coming to this conclusion, though, you should take a systematic approach.
- Based on the results, it is safe to conclude that the limit does not exist. The mathematical way to write this is:\[ \lim_{x \to 0} sin \left( \frac{1}{x} \right) \, DNE \]Where DNE stands for Does Not Exist.
- Of course, it is always a good idea to graph the function to confirm your result. The graph of \( f(x) = sin \left( \frac{1}{x} \right) \) shows that the function oscillates more and more wildly between \( -1 \) and \( 1 \) as \( x \) approaches \( 0 \).

There are times when saying that the limit of a function does not exist at a point does not provide enough information about that point. To see this, take another look at the second function from the beginning of this article.

\[ g(x) = \frac{|x-2|}{x-2} \]

As you choose values of \( x \) that are closer and closer to \( 2 \), \( g(x) \) does not approach a *single *value, but rather two values. Therefore, the limit does not exist, i.e.,

\[ \lim_{x \to 0} g(x) \, DNE. \]

While this statement is true, wouldn't you say that it doesn't quite give the full picture of the behavior of \( g(x) \) at \( x = 2 \)?

With **one-sided limits**, you can provide a more accurate description of the behavior of this function at \( x = 2 \).

For all values of \( x \) to the

**left**of \( 2 \) –**or the negative side of \( 2 \)**– \( g(x) = -1 \).So, you say that

**as \( x \) approaches \( 2 \) from the left, \( g(x) \) approaches \( -1 \)**. This is represented using mathematical notation as:\[ \lim_{x \to 2^{-}} g(x) = -1 \]

For all values of \( x \) to the

**right**of \( 2 \) –**or the positive side of \( 2 \)**– \( g(x) = 1 \).So, you say that

**as \( x \) approaches \( 2 \) from the right, \( g(x) \) approaches \( 1 \)**. This is represented using mathematical notation as:\[ \lim_{x \to 2^{+}} g(x) = 1 \]

Revisiting the third function from the beginning of this article, you will see there is a need to describe the behavior of functions that don't have finite limits.

\[ h(x) = \frac{1}{(x-2)^{2}} \]

From the graph of this function, you can see that as the values of \( x \) approach \( 2 \), the values of \( h(x) \) do not approach a value, but rather grow larger and larger, becoming infinite. This is represented using mathematical notation as:\[ \lim_{x \to 2^{+}} h(x) = +\infty \]

It is important to understand that when you say a limit is infinite, that does not mean the limit exists. It is simply a more descriptive way to say how the limit does not exist. \( \pm \infty \) is not a real number, so any infinite limit is not a limit that exists.

In general, limits at infinity are defined as:

__Three Types of Infinite Limits__

**Infinite limit from the left**: Let \( f(x) \) be a function defined at all values in an open interval \( (b, a) \).- If the values of \( f(x) \) increase without bound as the values of \( x \) (where \( x < a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) from the left is positive infinity. This is written as:\[ \lim_{x \to a^{-}} f(x) = +\infty. \]
- If the values of \( f(x) \) decrease without bound as the values of \( x \) (where \( x < a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) from the left is negative infinity. This is written as:\[ \lim_{x \to a^{-}} f(x) = -\infty. \]

**Infinite limit from the right**: Let \( f(x) \) be a function defined at all values in an open interval \( (a, c) \).- If the values of \( f(x) \) increase without bound as the values of \( x \) (where \( x > a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) from the right is positive infinity. This is written as:\[ \lim_{x \to a^{+}} f(x) = +\infty. \]
- If the values of \( f(x) \) decrease without bound as the values of \( x \) (where \( x > a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) from the right is negative infinity. This is written as:\[ \lim_{x \to a^{+}} f(x) = -\infty. \]

**Two-sided infinite limit**: Let \( f(x) \) be defined for all \( x \neq a \) in an open interval containing \( a \).- If the values of \( f(x) \) increase without bound as the values of \( x \) (where \( x \neq a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) is positive infinity. This is written as:\[ \lim_{x \to a} f(x) = +\infty. \]
- If the values of \( f(x) \) decrease without bound as the values of \( x \) (where \( x \neq a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) is negative infinity. This is written as:\[ \lim_{x \to a} f(x) = -\infty. \]

Use the limit laws to solve:

\[ \lim_{x \to -3} (4x+2) \]

**Solution**:

To solve this limit, apply the limit laws one at a time. Keep in mind that – at each step – you need to check that the limit exists before you apply the law. The new limit must exist for the law to be applied.

- Apply the sum law.\[ \lim_{x \to -3} (4x+2) = \lim_{x \to -3} 4x + \lim_{x \to -3} 2 \]
- Apply the constant multiple law.\[ \lim_{x \to -3} (4x+2) = 4 \cdot \lim_{x \to -3} x + \lim_{x \to -3} 2 \]
- Apply the basic limit.\[ \lim_{x \to -3} (4x+2) = 4 \cdot (-3) + 2\]
- Simplify.\[ \lim_{x \to -3} (4x+2) = -10\]

- Limits are all about determining how a function behaves as it approaches a specific point or value.
- The mathematical notation for a limit is:\[ \lim_{x \to a} f(x) = L \]
- Intuitively, limits can be evaluated using a table of functional values, or the graph of the function.
- There are several limit laws that make evaluating limits much easier:
- Two Important Limits\[ \begin{align}1. \; & \lim_{x \to a} x = a \\2. \; & \lim_{x \to a} c = c\end{align} \]
- Sum law for limits:\[ \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = L + M \]
- Difference law for limits:\[ \lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) = L - M \]
- Constant multiple law for limits:\[ \lim_{x \to a} (c \cdot f(x)) = c \cdot \lim_{x \to a} f(x) = cL \]
- Product law for limits:\[ \lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = L \cdot M \]
- Quotient law for limits:\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{L}{M} \mbox{ where } M \neq 0\]
- Power law for limits:\[ \lim_{x \to a} (f(x))^{n} = \left( \lim_{x \to a} f(x) \right)^{n} = L^{n} \mbox{ for every positive integer } n \]
- Root law for limits:\[ \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} = \sqrt[n]{L} \mbox{ for all } L \mbox{ if } n \mbox{ is odd, and for } L \geq 0 \mbox{ if } n \mbox{ is even} \]

In basic calculus, a limit is the value a function approaches as its input approaches some value.

To find the limit of a function, you directly substitute the value that the independent variable (usually x) is approaching, and solve.

If this is not possible, you can try some algebraic manipulation of the function, like

- factoring out common terms,
- multiplying a fraction by a conjugate,
- using trig transformations,
- looking at the graph of the function for limits at infinity,
- or using L'Hôpital's rule for indeterminate forms, like 0/0.

Once you simplify the limit using these methods, you can find the limit using direct substitution.

Put simply, a limit does not exist when the functional values do not approach a single value. The cases where this happens are:

- the function oscillates wildly as the limit is approached
- the one-sided limits are not equal
- when the limit is infinity (either from the left, right, or both sides)

More about Limits

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.