Limit Calculator

Enter a function and the calculator will determine its limits (Negative & Positive, One-tailed & Two-tailed). Get step wise solution for finite and infinite limit simplification with graph.

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Equation:

W.R.T:

Compute limit at (inf = ∞ , pi = π):

Direction:

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The limit calculator evaluates the limit of a function step-by-step as it approaches a specific value or infinity. It can analyze positive, negative, and infinity limits of any mathematical or calculus function, whether it has single or multiple variables.

What Are Limits In Math?

“Limit defines the behavior of a function at a certain point for any input change”

Limits of a function

Limit notation represents a mathematical concept that is based on the idea of closeness.

The calculator follows the same technique and 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.

Basic Limit Rules:

Limit calculator with steps works by analyzing various limit operations. These laws can be used to assess the limit of a polynomial or rational function manually as well.

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. Among these rules include:

Rules	Expressions
Sum/Difference Rule	lim_x→b[f(x) ± h(x)] = lim_x→b[f(x)] ± lim_x→b[h(x)]
Power Rule	lim_x→b[f(x)ⁿ] = [lim_x→b[f(x)]]ⁿ
Product Rule	lim_x→b[f(x) * h(x)] = lim_x→b[f(x)] * lim_x→b[h(x)]
Constant Rule	lim_x→b[k] = k
Quotient Rule	lim_x→b[f(x) / h(x)] = lim_x→b[f(x)] / lim_x→b[h(x)]
L'Hopital's Rule	lim_x→b[f(x) / h(x)] = lim_x→b[f'(x) /h'(x)]

How To Evaluate Limits?

Example # 01:

Evaluate the limit of the function below:

\(\lim_{x \to 3} 4x^{3}+6x{2}-x+3\)

Solution:

Here we will be using the substitution method:

Step 01:

Apply a limit to each and every value in the given function separately to simplify the solution:

\(= \lim_{x \to 3} \left(4x^{3}\right)+\lim_{x \to 3} \left(6x^{2}\right) - \lim_{x \to 3} \left(x\right) + \lim_{x \to 3} \left(3\right)\)

Step 02:

Now write down each coefficient as a multiple of the separate limit functions:

\(= 4 * \lim_{x \to 3} \left(x^{3}\right)+6 * \lim_{x \to 3} \left(x^{2}\right) - \lim_{x \to 3} \left(x\right) + \lim_{x \to 3} \left(3\right)\)

Step 03:

Substitute the given limit i.e;

\(\lim_{x \to 3}\):

\(\lim_{x \to 3} 4x^{3}+6x{2}-x+3 = 4 * \left(3^{3}\right) + 6 * \left(3^{2}\right) - 3 + 3\)

Step 04:

Simplify to get the final answer:

\(\lim_{x \to 3} 4x^{3}+6x{2}-x+3 = 4 * 27 + 6 * 9 - 3 + 3\)

\(\lim_{x \to 3} 4x^{3}+6x{2}-x+3 = 108 + 6 * 9 - 3 + 3\)

\(\lim_{x \to 3} 4x^{3}+6x{2}-x+3 = 162\)

Example # 02:

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

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.

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

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

Moving ahead further now:

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

\(= \frac{cos 0}{1}\)

\(= 1\)

How Does Limit Calculator Function?

The tool is straightforward to use! It requires a few inputs to calculate limits of the given function at any point that include:

Inputs To Enter:

Enter the 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
Tap Calculate

Results You Get: