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Limit Calculator

Limit Calculator

Enter any function and the limit calculator will immediately calculate its limits (positive and negative), with step-by-step calculations shown.



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Compute limit at: (inf = ∞ , pi = π)


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This series limit calculator computes positive or negative limits for a given function at any point. You must give a try to this limit solver to determine how to solve limits with ease. Also, this l’hopital’s rule calculator helps to calculate \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\) limit problems and supports computing limits at positive and negative infinities.

Well, read on to gain insight about how to find the limit of a function by using this limit evaluator. Let’s begin with some basics!

What Is A Limit Math?

It can be defined as:

“The value that a function “approaches” as the input “approach” some value.”

Limit notation represents a mathematical concept that is based on the idea of closeness. It is necessary to evaluate the limit in calculus either by calculus limit calculator or by hand. The limit definition calculator assigns values to certain functions at points where no values are defined. It does this all in such a way as to be consistent with proximate or near values.

In most calculus courses, we work with a limit that means it’s easy to start thinking the calculus limit always exists. On the other hand, it also helps to solve the limit by l’Hopital’s rule, which would be explained in the read below.

Well, an online derivative calculator is the best way to calculate the derivative of the function by given values and shows differentiation, which is an important aspect of l’hopital calculator in finding the limits.

Limit Equation:

Now you could easily calculate limits of any functions by using the formula that is also used by limits calculator and is given below:

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

Example Description:

If you have a function \(\frac {(x2 − 1)} {(x − 1)}\), then there’s a need to finding limits when x is 1 as division by zero is not a lawful mathematical operation. On the other hand, for any other value of x the numerator can be factored in as well as divided by the (x-1), to give x+1. Thus, this quotient will be equal to x+1 for all values of x excluding 1, which has no value. Though, 2 can be assigned to the function \(\frac {(x2 − 1)} {(x − 1)}\) as its limit when x approaches 1. If the limit of x approaches 0 or infinity such calculations can be made easier by the use of l’hopital’s rule calculator with steps.

For solving limits, there are certain laws and limit calculators available that use calculus rules to determine the limit of a function. Also, the free online integral calculator enables you to determine the integrals of a function corresponding to the variable involved and show you the step-by-step work.

The Limit Laws:

This best lim calculator with steps works by analysing various limit operations. These laws can be used to assess the limit of a polynomial or rational function. Additionally, there are certain conditions for some rules and if they are not satisfied, then the rule cannot be used to validate the evaluation of a limit. However, using a limit evaluator is the best way to evaluate the limits of a function at any point.

The following table summarises the limit laws along with some central properties.

  Limit Law in symbols Limit Law in words
1  \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x)\) The sum of Limits is equal to limit of a sum.
2  \(\lim_{x \to a}[ f(x) – g(x)]= \lim_{x \to a} f(x) – \lim_{x \to a} g(x)\) Difference of limits is equal to limit of difference.
3 \( \lim_{x \to a} cf (x) = c \lim_{x \to a} f (x) \) Constant times the limit of function is equal to limit of a constant times of a function.
4  \(\lim_{x \to a}[ f(x)g(x)] = \lim_{x \to a} f(x) × \lim_{x \to a} g(x)]\) The product of the limits is equal to limit of a product.



\(\lim_{x \to a} \frac {f(x)} {g(x)} = \frac {\lim_{x \to a} f(x)} {\lim_{x \to a} g(x) }\) The quotient of the limits is equal to limit of a quotient.
6 \(\lim_{x \to a}[ f (x)]^n = [\lim_{x \to a} f (x)]^n\) Where the value of \(n\) is a positive integer.
7  \(\lim_{x \to a}c = c\) The constant is equal to the limit of a constant function.
8 \(\lim_{x \to a}x = a\) The limit of a linear function is equal to the number \(x\) is approaching.


9 \(\lim_{x \to a} x^n= a^n\) The limit where the value of \(n \)is a positive integer.
10 \( \lim_{x \to a} x ^ n = a ^ n\) The limit where value of \(n \)is a positive integer & if \(n\) is even.
11 \(\lim_{x \to a} f (x)^n =  lim_{x \to a} f (x)^n \) Where the value of \(n\) is a positive integer & if \(n \)is even.

How To Evaluate Limits?

Various ways are out there to find limits, one of best which is the free l’hopital’s calculator. But here we will be elaborating each and every one of them one by one. Let’s go!

Substitution Method:

In this method, we are required to substitute the value of limit in the function to evaluate the limit. You can also do this swiftly by subjecting to this online evaluate limit calculator. How does it sound to you?

Factoring Method:

In this method, you need to make factors so that like terms may be cancelled out. And when remaining terms are there, you have to apply limits on them as per the rule.

L’Hopital’s Rule:

This is indeed one the most important techniques used to evaluate the limits that are in the form of \(\frac{0}{0}\) or \(\frac{\inf}{\inf}\). To apply this method, you need to consider the following points in mind:

  • First, follow the substitute method to evaluate the limit. If it results \(\frac{0}{0}\) or \(\frac{\inf}{\inf}\), then you are required to apply this rule. Another ease here is that this free l hospital rule calculator takes a couple of moments in recognizing whether the given function is undefined or not.
  • Now when you get the form as defined above, you need to differentiate the function
    After that, you have to apply the substitution method to find the limit of the given function.

For better approximations, we will get to know how to find limits by using this particular way in example later on.

Conjugate Method:

In this method, we mostly encounter complex numbers. Now what our advice here is to determine the conjugate of the number given. After you do so, you need to simplify it and use this infinite limit calculator to resolve boundaries of any function.

Infinite Limit and Rational Function:

A function that can be written as the ratio of two polynomials:

$$ f(x) = \frac {P(y)} {Q(y)} $$

We can find the function’s limit is 0, Inf, -Inf, or calculated by coefficients.

How To Calculate Limits?

What about resolving a few examples to comprehend how you make use of the various methods to simplify limits? Let’s dive into it!

Example # 01:

Evaluate the limit of the function below:

$$ \lim_{x \to 13} \frac {x} {5} $$


Here we will be using the substitution method:

$$ \lim_{x \to 13} \frac {x} {5} $$

Put x = 3;

$$ \lim_{x \to 13} \frac {x} {5} = \frac{13}{5} $$

$$ \lim_{x \to 13} \frac {x} {5} = 2.6 $$

Which is the required answer.

Example # 02:

Calculate the limits of the function given as follows:

$$ \lim_{y \to 2} \frac {y^2} {y-2} $$


Here by factoring \((y^2 – 2^2 )\), we will get \(\left(x+2\right)\) and \(\left(x-2\right)\)

So we have:

$$ \lim_{y \to 2} \frac {y^2 – 4 } { y – 2} $$

$$ = \lim_{y \to 2} \frac {( y-2 ) ( y + 2 )} {( y-2 )} $$

Now substituting y = 2;

$$ \lim_{y \to 2} ( y + 2 ) $$

$$ 2+2 = 4 $$

You can also cross check all the answers by using this best find the limit calculator in a blink of moments.

Example # 03:

How to do limits for the function given as below:

$$ \lim_{x \to 0} \left(\frac{sin x}{x}\right) $$


Using the substitution method:

$$ \lim_{x \to 0} \left(\frac{sin x}{x}\right) $$

$$ = \frac{sin 0}{0} $$

$$ = \frac{0}{0} $$

Which is an indeterminate form. So here we will be applying l’hopital’s rule:

Before we move on, we have to check whether both the functions above and below the vinculum are differentiable or not.

$$ \frac{d}{dx} \left(sin x\right) = cos x $$

$$ \frac{d}{dx} \left(x\right) = 1 $$

Moving ahead further now:

$$ \lim_{x \to 0} \left(\frac{cos x}{1}\right) $$

$$ = \frac{cos 0}{1} $$

$$ = 1 $$

To speed up your calculations, try using the l’Hospital rule calculator.

Example # 04:

How to find the limit of the following function:

$$ \lim_{z \to 9} \frac {3 – \sqrt {z}} {9 – \sqrt {z}} $$


If the value of z is equal to 9 put in the equation it gives 0/0, which is not the right answer.

So, let’s began with rearrange:

$$ = \frac {3 – \sqrt {z}} {9 – \sqrt {z}} * \frac {3 – \sqrt {z}} {3 – \sqrt {z}} $$

Multiplying top and bottom by conjugate of top:

$$ = \frac {3^2 – \sqrt {z}^2} {( 9 – z ) (3 + \sqrt {z})} $$

$$ = \frac {( 9 – z )} {( 9 – z ) (3 + \sqrt {z})} $$

$$ = \frac {1} { 3 + \sqrt {z} } $$

$$ \lim_{z \to 9} \frac { 3 + \sqrt {z} } { 9 – z } $$

After cancelling 9-z;

$$ \lim_{z \to 9} \frac {1} { 3 + \sqrt {z} } $$

$$ = \frac {1} { 3 + \sqrt {9} } $$

$$ = \frac {1} { 3 + 3 } $$

$$ = \frac {1} {6} $$

How Limit Calculator Solve Limits?

This limit graphing calculator allows you to evaluate the limit of the given variables. Well, the limit finder assists to find the limits by following the given steps:


  • First of all, enter the equation or function.
  • Select the variable from the drop-down with respect to which you need to evaluate the limit. It can be x,y,z,a,b,c, or n.
  • Specify the number at which you want to calculate the limit. In this field, you can use a simple expression as well such as inf=∞ or pi =π.
  • Now select the direction of the limit. It can be either positive or negative
  • Once you enter the values in the given fields, the calculator will give you an equation preview.
  • Hit the calculate button.


The free limit evaluation calculator does the following calculations:

  • Determine the limits of the given function
  • Display the step by step calculations
  • Represents the Taylor’s series expansion for the given function


How do you know if a limit does not exist?

To find a limit on a graph if there is a vertical asymptote and one side goes toward infinity and the other goes in the direction of negative infinity, then the limit does not exist. In the same way, if the graph has a hole at the x value c, then the two-sided limit won’t exist. However, a limit finder can help you out in assessing the limits more accurately.

What is correct limit notation?

Basically, a limit notation is a way of stating a delicate idea than simply saying x=5 or y=3 \(\lim_{x \to a} f(x)=b\) On the other hand, a one sided limit calculator takes away the worry of limit notation as it solves limits and states them inaccurate formatting.

Can L‘Hopital’s rule be applied to every limit?

L’Hôpital’s Rule is used with unspecified limits that have the form 0/0 or infinity. It doesn’t solve all kinds of limits. At times, even recurrent applications of the rule cannot help to find the limit values. So, for convenience, a l’Hopital’s rule calculator is the best way to solve infinite limits of functions.

Can 0 be a limit?

If we are simply evaluating the equation 0/0 limit will be undefined. However, if we get 0/0 then there might be a series of answers. Now the only way to determine the accurate answer is to use a limit solver to determine the limit problems accurately.

How are limits used in calculus?

Limits define how a function will act near a point, as an alternative of at that point. This idea is the basis of calculus. For example, the limit of “f” at x=3 is the value f as we get closer and closer to x=3. That could also be obtained by subjecting this to the best limits calculator.


This online Limit calculator finds limits and specifically functions to solve limits with respect to a variable. Limits can be evaluated on either positive or negative sides. It caters to all the limit problems that are impossible to do algebraically. Thus it is great to help students and professionals to solve and verify your limits in the blink of an eye.


From the authorized source of Wikipedia: Limit (mathematics), function, sequence, standard parts and much more!

The source of Khan Academy provides with: Best Strategy in finding limits

From the source of tutorial.math: All you need to know about the limit approaches

Other Languages: Limit Hesaplama, Kalkulator Limit, Grenzwertrechner, Kalkulačka Limit, Calculadora De Limites, Calculateur De Limite, Calculadora De Limites, Calcolatore Limiti, Калькулятор Пределов.