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Finding Limits

Finding Limits

Like beads on a string leading to a pendant, points on a graph can lead you to the limit of a function. How can we use points on a graph to evaluate limits? Good question! Here we look at some of the different ways of finding limits of functions!

Finding Limits in Calculus

There are lots of ways to find the limit of a function!

  • You can use the , definition of the limit and write a proof. See Limits of a Function for examples of this technique.

  • You can look at the graph or a table of values to see what the limit might be. See Finding Limits Using a Graph or Table for plenty of examples of finding limits this way.

  • You could look at the limit from the left and the limit from the right of a function, and see if they are the same. See One-Sided Limits for definitions and examples of using this technique.

  • You could use Limit Laws, which are theorems that have already been proven to find the limit. If your function is nice this is often the way people find the limit. For more information on properties of limits see Limit Laws

  • You may need to use a special theorem to find the limit, like the Squeeze Theorem or the Intermediate Value Theorem. Both of them are very useful and the Intermediate Value Theorem will turn up later in topics like finding the maximum value of a function. See The Squeeze Theorem or see The Intermediate Value Theorem for details on how to use them.

Here you will see a sampling of the ways to find the limit of a function.

Using the Definition of the Limit

To review the definition of the limit of a function, see Limits of a Function.

Take where and are constant real numbers. Is it true that

?

Answer:

Yes. Using the definition, for any you are given,

no matter what you use. So constant functions have the limit you would expect them to.

Take , and let be a constant real number. How do you know that

?

Answer:

You might be tempted to say "of course, the limit is - the function is just a line". In fact, that is almost enough. You can't use any of the Properties of Limits, but you can use the definition and take to show that the limit is .

Using the Rules for Finding Limits

For a review of the various properties of limits and how to use them, see Limit Laws.

Take the function , and to be a constant real number. Find

.

Answer:

Notice that the function is just the sum and product of powers of along with the constant . You already know that

and

from the two examples above, which means the conditions to apply the Sum Rule, Product Rule, and Constant Rule are met. Then applying them gives

Finding Limits Graphically

Below is an example of using the graph to find the limit of a function. For more information on problems like this see Finding Limits Using a Graph or Table.

Consider the function

,

Find the limit of the function as .

Answer:

First, graph the function and make a table of values near . Although the function has more roots than are shown in the graph, since you only care about the limit as , it makes sense to zoom in on the function there.

Finding Limits example using graph and table StudySmarterUsing a graph and table to find the limit of a function | StudySmarter Original

The points on the graph correspond to the points in the table. You can see from both the graph and table that as gets closer and closer to , the function values get closer and closer to That means that

.

Notice that you don't actually care about the function value at when finding the limit, because the definition says to look close to but not at .

Finding Limits Algebraically

There are more examples of finding limits algebraically in a separate article. See Finding Limits of Specific Functions.

In fact, limits and continuity also go together.

If a function is continuous at a point, then the limit of the function exists and is equal to the function value at that point.

From the previous example, we had

,

and found the limit as . Since you know that all polynomials are continuous everywhere (see Continuity and see Theorems of Continuity for more details), you know the limit of the function exists and is equal to the function value. Since , that means

.

Look at the function

and find the limit as .

Answer:

The function is undefined at , so you can't just plug in the function value to find the limit. But you can factor the numerator to get

as long as . That means the graph of the function is actually the straight line with a hole at the point . So

Finding the Derivative Using the Limit Definition

Using the definition of the derivative does involve limits. This is a big topic and it has a whole article on its own! See our article on Derivatives for lots more details on how to find the derivative using a limit.

Finding Limits - Key takeaways

  • For any polynomial , .
  • A table or graph can be used to find the limit of a function.
  • Finding limits algebraically can involve factoring the numerator and denominator and seeing if anything cancels out. This is especially useful in cases where there is a hole in the graph.
  • Properties of Limits can also be used to take the limit of functions.

Frequently Asked Questions about Finding Limits

There are lots of techniques for finding limits, including using the definition, using a graph or table, using properties and theorems of limits, or algebraically.

Look at function values close to the points you are taking the limit at.  If the function values all get close to the same number, then that is the limit.

There are lots of techniques for finding limits, including using the definition, using a graph or table, using properties and theorems of limits, or algebraically.

There are many algebraic techniques for finding the limit, including factoring and using continuity.

Look at the sequence values as n gets very large.  If the sequence values all get close to a single number, then that number is the limit of the sequence.

Final Finding Limits Quiz

Question

Which one of these is not a way to find the limit of a function

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Answer

Letting your dog pick the answer.

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Question

Which one of these is not a way to find the limit of a function.

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Answer

Asking your cat for advice.

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Question

If a function is continuous at a point, does it have a limit there?

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Answer

Yes, and in fact the limit is equal to the function value there.

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Question

True or False:  When finding the limit of a function at a point, the value of the function at that point matters.

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Answer

False.  You care about function values near the point, but not at the point.

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Question

Do limits of functions always exist?

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Answer

No.  Only if the limit of the function is a number do you say it exists.  There are lots of function that have vertical asymptotes, and at those places the function doesn't have a limit.

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Question

True or False:  You can always use the definition to find the limit of a function.

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Answer

Well, technically this is true, but it can often be quite challenging.  That is why we have things like Limit Laws and the Squeeze Theorem.

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Question

Is it true that polynomial functions have limits everywhere?

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Answer

Yes.  Since polynomials are continuous everywhere, they have a limit everywhere and it is equal to the function value.

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Question

Can you always use the Quotient Rule to find the limit of a rational function?

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Answer

No, you can only use it in the case when the limit of the denominator is not zero.

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Question

What is one of the things you can do to try find the limit of a rational function?

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Answer

You could try factoring and cancelling, or doing algebra to make things easier, or using properties of limits.

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Question

If you are trying to find the limit of a function which has a root in it, what is one of the techniques you can use?

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Answer

You can try multiplying both numerator and denominator by the conjugate of the part with the root.

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Question

If you want to find the limit of a piecewise defined function at the point where the function definition changes, what do you need to do?

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Answer

Find the limit from the left there, and the limit from the right there, and see if they are the same.  If they aren't the same then the limit doesn't exist.

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Question

How do you find the limit of an exponential function?

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Answer

You can think of it as the composition of two functions, and then use the fact that the exponential function is continuous to find the limit.

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Question

Define a limit of a function.

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Answer

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:

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

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Question

By which of the following ways one can find limit?

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Answer

Using tables.

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Question

How do you find a limit using table?

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Answer

  • Make a table using \(x\) and \(f(x)\) values with \(x\) approaching \(a\) from both sides.
  • If you get the same answer from both the left and the right, then that is the limit.

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Question

Does one side limit exist?

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Answer

Yes.

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Question

Is it possible to identify the limit with a function approaching \(\infty\) or \(-\infty\) in a graph?

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Answer

Yes

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Question

The vertical line \(x=a\) is called ____ for function approaching \(\infty\) or \(-\infty\).

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Answer

Asymptote.

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Question

What is the limit for the given function?

\[\lim\limits_{x \to 0}\frac{x^2-25 }{x^2-4x-5}.\]


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Answer

5.

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Question

If for a function, both the right-hand limit and left-hand limit have different values then the limit of the function  _____?

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Answer

Is the one which is closer to 0.

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Question

The right-hand limit of a function approaches from ____.

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Answer

Right side.

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Question

Find the limit for \[\lim\limits_{x \to \infty }\cos\left ( \frac{1}{n} \right ).\]

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Answer

\(\lim\limits_{x \to \infty }\cos\left ( \frac{1}{n} \right ) = 1\).

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Question

Can you check the limit of a function in a graph without creating a table?

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Answer

Yes.

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