Limit Calculator With Free Steps

Limit Calculator

Enter a function, provide the value at which you want to calculate the limit, and click the 'Calculate' button to find its limit.

Limit Calculator

The Limit Calculator evaluates the limit of a function step-by-step as it approaches a specific value or infinity. It can handle positive, negative, and infinity limits for any mathematical or calculus function, including single or multiple variables.

What Are Limits in Math?

"A limit defines the behavior of a function at a certain point as its input changes."

Limits of a function

Limits capture the idea of "closeness" and allow the evaluation of functions at points where they may not be directly defined. The calculator follows this principle to provide consistent results.

Basic Limit Rules

The limit calculator uses the following rules to simplify calculations. These can also be applied manually to polynomials or rational functions:

Rule	Expression
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’Hôpital’s Rule	lim_x→b[f(x)/h(x)] = lim_x→b[f'(x)/h'(x)]

How to Evaluate Limits?

Example 1:

Evaluate the limit:

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

Solution:

Apply the limit to each term separately:
\(\lim_{x \to 3} 4x^3 + \lim_{x \to 3} 6x^2 - \lim_{x \to 3} x + \lim_{x \to 3} 3\)
Factor out constants:
\(4 \cdot \lim_{x \to 3} x^3 + 6 \cdot \lim_{x \to 3} x^2 - \lim_{x \to 3} x + \lim_{x \to 3} 3\)
Substitute x = 3:
\(4(3^3) + 6(3^2) - 3 + 3\)
Simplify:
\(108 + 54 - 3 + 3 = 162\)

Example 2:

Evaluate the limit:

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

Solution:

Direct substitution gives 0/0, an indeterminate form. Use L’Hôpital’s Rule:

\(\frac{d}{dx} (\sin x) = \cos x\)
\(\frac{d}{dx} (x) = 1\)

Apply L’Hôpital’s Rule:

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

How the Limit Calculator Works

Inputs:

Enter the function.
Select the variable (x, y, z, a, b, c, or n).
Specify the point to evaluate the limit (use expressions like ∞ or π if needed).
Choose the direction (positive or negative).
Click Calculate.

Outputs:

Step-by-step calculation of the limit.
Final evaluated limit value.
Optional: Taylor series expansion of the function near the point.

References

Wikipedia: Limit (mathematics)
Khan Academy: Strategies for Finding Limits

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EN	TR	ID
DE	CS	PT
FR	ES	IT
RU