# Limits Calculator

Limits Calculator calculates the limit of the given function at a certain point. Limit is defined as the value that the function attains as the input approaches a mentioned number. Limits are used to analyze the behavior of a given function.

## What is Limits Calculator?

Limits Calculator is an online tool that helps to calculate the value of the function as the input approaches the given point. When we want to make approximations while performing calculations, we use limits. These help to determine the value of a quantity as close as possible to its actual value. To use this * limits calculator*, enter the values in the given input boxes.

### Limits Calculator

## How to Use Limits Calculator?

Please follow the steps below to find the limit of the function using the online limits calculator:

**Step 1:**Go to Cuemath’s online limits calculator.**Step 2:**Enter the function and the limit value in the given input boxes of the limits calculator.**Step 3:**Click on the**"Calculate"**button to find the limit of the function.**Step 4:**Click on the**"Reset"**button to clear the fields and enter new values.

## How Does Limits Calculator Work?

Say we have a function y = f(x). Let us suppose f(x) takes an indeterminate form at x = a. We consider the values of the function that are close to a. If these values tend to some unique number as x approaches a then we can say that this unique number is a limit of the function f(x) at x = a. The limits formula can be given by:

\(\lim_{x\rightarrow a}f(x) = A\)

There are many different methods to evaluate limits. Some of them are given below.

**Direct Substitution**- We can obtain the limit of a continuous function by direct substitution. Most polynomial function limits can be determined by this method. We need to substitute the value of the variable in the given function to get the answer.**Factorization**- Suppose we have a function that results in an indeterminate form (e.g., 0/0) on direct substitution. We need a different procedure to solve these limits. In the factorization method, we break the denominator and numerator into factors. On cancellation of the common factors, the expression reduces to a determinate form. This can be easily solved by substituting the values of the variables.**Rationalization**- We can rationalize an indeterminate expression to get a determinate form. Finally, this can be solved by substituting the variable values.

## Solved Examples on Limits

**Example 1:** Find the value of limits \(\lim_{x\rightarrow 2}x^{2} + 3x + 5\) and verify it using the limits calculator.

**Solution:**

\(\lim_{x\rightarrow 2}x^{2} + 3x + 5\)

Substitute the limit value 2 in given function f(x)

= 2^{2} + 3(2) + 5

= 4 + 6 + 5

= 15

Therefore, the value is 15.

**Example 2:** Find the value of limits \(\lim_{x\rightarrow -4}x^{3} + 4x^{2} + 7\) and verify it using the limits calculator.

**Solution:**

\(\lim_{x\rightarrow -4}x^{3} + 4x^{2} + 7\)

Substitute the limit value 4 in given function f(x)

= (-4)^{3} + 4(-4)^{2} + 7

= 7

Therefore, the value is 7.

Similarly, you can use the limits calculator to find the value of limits for the following:

- \(\lim_{x\rightarrow -2}x^{4} - 3x^{2} + 1\)
- \(\lim_{x\rightarrow 5}x + 1\)

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