How to Evaluate Limits? Complete Guide [Easy Explanation]

How to Evaluate Limits

Limit, it is the value that a function approaches, when the input approaches a specific value. Let us see the formula for limit which will help us to Evaluate Limits.

We say that the limit f of x  is “L” as x approaches ‘a’ and write this as

provided we can make f(x) as close to L as we want for all x sufficiently close to a.

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Let’s Understand How to Evaluate Limits

Let us take a straightforward example and see how limits make a problem easier.


Now let try to solve the problem given above

By using the trigonometric identity, the above-given equation can also be as follows,

Now let us assume 3 = x therefore, 3d = dx

To solve this further, we know trigonometric values

Which are when,

And when,

Now keeping these trigonometric values in mind, we should proceed further.

Now we are going to substitute the values of trigonometric terms in the above equation.

After substituting the value, you can perform a cancellation operation to solve the equation further and get the final answer.

4 Simple Ways to Evaluate Limits

Evaluating means to find the value of something that is mentioned in the question.

In the example above we said the limit was 2 because it looked like it was going to be the exact value, but there is much more to it!

There are many ways to get an accurate answer by limits. Let’s look at some:

Refer: What is the Fourier series? Easily Explained with Examples

Substitute Method

The first thing for evaluating such limits is try to substitute the value of the limit in, and see if it works.


Let us see another example given below:

Factors Method

We can try this method as well for Evaluating Limits




By factoring (x^2−1) into (x−1)(x+1) we get:



Now we can just put x=1 to get the limit:



Conjugate Method

For some questions to evaluate limits, multiplying both top and bottom by a conjugate will do the work.

The conjugate is where we perform a sign change operation in the middle of two terms or more. Let us understand it with this example mentioned below.

Example:    3x + 1 becomes 3x – 1.

In the below-mentioned example where it will help us find a limit:

Evaluating this at x=4 gives 0/0, which is not a good answer!


So, let’s try some rearranging:

Multiply top & bottom by the conjugate of the top: Simplify top using,

Simplify top further,

Cancel (4−x) from top and bottom:

So, now we have:

We just found out the answer.

Infinite Limits and Rational Functions Method

Rational Function is basically the ratio of two polynomials functions:


By finding the Degree of the function we can find out whether the function’s limit is zero, Infinity, -Infinity, or any other possible outcome.


Calculus is a little difficult topic of mathematics as compared to others, but when it comes to evaluate limits, you can be relaxed. Limits are a bonus to calculus but not in terms of difficulty. There are a variety of questions available under the limits segment you have to practice them to master them all.

In this article, we discussed how to Evaluate limits, and we saw some easy examples. We also saw four different methods for evaluating limits and saw examples of them. To gain expertise in it, you have to do more and more practice for it. We believe that this article helps you with the understanding of the concepts of limit evaluation.

Now go ahead start from the very basics and keep on practicing question. Always remember to keep the level of your questions progressive, starting from basic to advance.

All the best!

Also Refer: What is Continuous Integration?


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