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# Limit Calculator

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

Equation:

W.R.T

Compute limit at: (inf = ∞ , pi = π)

Direction:

Equation Preview

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This 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.

## 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.

### 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. 5 $$\lim_{x \to a} \frac {f(x)} {g(x)} = \frac {\lim_{x \to a} f(x)} {\lim_{x \to a} g(x) }$$ 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?

What about resolving a few examples to comprehend how you make use of the various methods to simplify limits?

Example # 01:

Evaluate the limit of the function below:

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

Solution:

Here we will be using the substitution method:

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

Put x = 3;

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

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

Example # 02:

Calculate the limits of the function given as follows:

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

Solution:

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:

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

Solution:

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.

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

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

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

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

$$= 1$$

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

Example # 04:

How to find the limit of the following function:

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

Solution:

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:

Input:

• 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.

Output:

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

## References:

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