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Limit Laws

You've studied the definition of limits and how to find them algebraically and graphically. Is there perhaps a faster way to find the limit of a function. Yes, there is! We can use limit laws. Here you will see some of the more common properties of limits of functions and how to apply them.

Importance of Limit Laws

You may ask yourself why limit laws in calculus are important. You already know the definition of the limit of a function. Why not just apply that? The reason is that it is much more efficient to prove one thing about functions in general than to use the definition on each and every function. It is the difference between proving that dogs like to play and proving that my dog likes to play, your dog likes to play, the neighbor's dog likes to play... and on and on and on.

Limit Laws in Calculus

Many textbooks will mention the properties of limits listed below since they are the most common ones. Sometimes they will even refer to them as the 5 limit laws.

Theorem: Properties of Limits, also known as Limit Laws

Suppose that , , and are real numbers, with and being functions such that

and .

Then the following hold:

Sum Rule:

Difference Rule:

Product Rule:

Constant Multiple Rule:

Quotient Rule: If then

Power Rule: If , with , then

provided that is a real number and when is even.

For more examples of how to find limits of particular functions, see Finding Limits. For a reminder on the definition of the limit of a function, see Limits of a Function.

It is essential to make sure that the conditions are met before applying the properties of limits. Let's see an example.

Take

and ,

and try to find

.

You are probably tempted just to use the Product Rule for limits. You already know that

.

However, if you try and apply the definition of the limit for , you can see that no matter how close you take your window to be to , it won't work because the function has a vertical asymptote at . So, doesn't have a limit at . But

which is not what you get when you multiply together and something that doesn't exist! So while the limit of the product does exist, the product of the limits does not.

Calculating Limits Using the Limit Laws

For some functions, the limit laws get used so much that it is easier to look at kinds of functions rather than at lots of functions. It turns out that polynomials and rational functions are especially nice.

Definitions and Limit Laws

In the following examples, the definition of the limit was used to show that

and

where is a constant. See Limits of a Function for more details on how to apply the definition of the limit.

Take the function

,

and to be a constant real number. Find

.

Looking carefully, you can notice that the function is just the sum and product of powers of , along with the constant! So, the condition required to use our limit laws is met! Applying them gives:

In the example above, you looked at a specific polynomial and found the limit exists. It turns out that you can do this same process (using the Sum Rule, Constant Rule, and the Power Rule) to find the limit of any polynomial!

If is a polynomial and is a real number, then

.

Limits of Rational Functions

Taking the limit of rational functions can sometimes be a challenge. For more examples of techniques you can use for rational functions, see Finding Limits of Specific Functions.

In the case where the point you want to take the limit at is in the domain of the rational function, taking the limit is not difficult. Take

and find the limit as .

First, ask yourself if is in the domain of the function. It turns out that it is, and in fact

.

So using the Quotient Rule tells you that

.

Similar to the result for polynomials, you can say the following about rational functions:

If is a rational function and is a real number in the domain of , then

.

Examples Using the Limit Laws

Rather than looking at a function with a definition, sometimes all you will know are some properties of the functions involved, and you will need to use Limit Laws to draw conclusions about the functions.

Suppose that

, and .

If possible, find the following:

,

,

,

and

.

1. To find

you have all the conditions satisfied to apply the Sum Rule, so

.

2. You can use the Constant Rule for the next one, so

.

3. Since the limit of as is not equal to zero, you can apply the Quotient Rule to see that

.

4. The last one is a bit more challenging. Here,

,

You cannot take the square root of a negative number and get a real number back, so you can't evaluate this. That means you can't find

.

Limit Laws - Key takeaways

• Suppose that , , and are real numbers, with and being functions such that

and .

Then the following hold:

Sum Rule:

Difference Rule:

Product Rule:

Constant Multiple Rule:

Quotient Rule: If then

Power Rule: If , with , then

provided that is a real number and when is even.

• If is a polynomial and is a real number, then

.

• If is a rational function and is a real number in the domain of , then

.

• Always be sure that the conditions to use one of the Limit Laws are met before you use it!

There are actually a lot more than 5 theorems about limits, but you probably mean the sum/difference rule, the constant multiple rule, the product rule, the quotient rule, and the power rule.

Experimentation and research.

By using the fact that both functions have a limit which is an actual number.

They are really properties of limits that you can use so you don't have to show each and every different function has a limit.

Lots!  Most textbooks will talk about 5 main ones, but there are actually more of them.

Final Limit Laws Quiz

Question

Summarize the Squeeze Theorem in one sentence.

The Squeeze Theorem is a limit evaluation method where we "squeeze" an indeterminate limit between two simpler ones; the "squeezed" function approaches the same limit as the other two functions surrounding it

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Question

What is the step-by-step procedure for the Squeeze Theorem?

• Step 1: Create double-sided inequality based on the nature of f(x)
• Step 2: Algebraically modify the inequality as needed
• Step 3: Solve the limits on both sides of the inequality (they must be equal to continue)
• Step 4: Apply the Squeeze Theorem

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Question

Should you always try the Squeeze Theorem first when solving limits?

No! The Squeeze Theorem is a last resort method and only should be used if algebraic manipulation fails

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Question

The requirements of the IVT are:

• f(x) must be a continuous function on the closed interval
• on the interval [a, b], f(a) < f(c) < f(b)

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Question

Why is the Intermediate Value Theorem important?

• it is a fundamental theorem in Mathematics that is used to prove a variety of other theorems
• useful in proving a function has a solution

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Question

What should you do first before trying to apply a limit law?

Check that all the functions are defined at the point where you want to find the limit at!

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Question

What is another name for the Squeeze Theorem?

Sandwich Theorem

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Question

Which family of functions are particularly good candidates for the Squeeze Theorem?

trigonometric functions

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Question

What is the first method you should try for solving limits?

You should always try algebraic or simple manipulation methods first.

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Question

Why is the Squeeze Theorem called the Squeeze Theorem?

A function is "squeezed" between two simpler functions

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Question

What theorems is the IVT used to prove?

Extreme Value Theorem and the Mean Value Theorem

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Question

If a function meets the Intermediate Value Theorem requirements,  ______ solution is/are guaranteed.

one

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Question

What important question in mathematics does the Intermediate Value Theorem answer?

Does an equation have a solution?

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