# Limit L'hopital's Rule Calculator

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## L’hopital’s Rule Calculator

Find the values of the undetermined limits using L’hopital’s Rule Calculator. This tool will differentiate the function and apply the limit to find its value.

## How to use this tool?

The steps to use the L’hopital’s Rule calculator are as follows.

• Enter the functions. Add any math keys by clicking on the keyboard icon.
• Select the variable and the side of the limit.
• Lastly, enter the value of the limit.
• Click Calculate to find the answer.

## What is the L’hopital’s Rule?

L’hopital’s rule is a technique used in calculus to find the value of undetermined forms like 0/0 or ∞/∞.

An indeterminate form is a mathematical expression that doesn't have a clear and direct value. Common examples include 0/0, ∞/∞, 0 x ∞, - , 00, and ∞0. An indeterminate form doesn't imply that the limit doesn't exist or that it doesn't have a specific value.

Instead, it means that the limit cannot be directly determined in its current form, and additional steps are required to find its value. For example, a limit that results in the form 0/0 is indeterminate because we cannot directly conclude its value from the form itself.

## Formula of the L’hopital’s rule:

Suppose f(x) and g(x) are functions that are differentiable near a point a (except possibly at a) and these three conditions are met:

1. lim x→a f(x) = 0 or lim x→a f(x) = ∞
2. lim x→a g(x)=0 or lim x→a g(x) = ∞
3. lim x→a f’(x)/g’(x) exits or is equal to ∞.

This means that

lim x→a f’(x)/g’(x) = lim x→a f(x)/g(x)

## How to use the L’hopital’s rule?

To apply L'Hôpital's rule, follow these steps:

1. Verify that the limit is in the form 0/0 or ∞/∞.
2. Differentiate the numerator and denominator separately.
3. Calculate the limit of the quotient of the derivatives.
4. If the limit obtained in step 3 is still indeterminate, apply L'Hôpital's rule again until you find a determinate form.

Example 1:

Find the limit of $\lim_{x \to \infty} \frac{e^x}{x^2}$.

Direct Substitution:

If we substitute x=∞ into the expression, we get

$\lim_{{x \to \infty}} \frac{{e^x}}{{x^2}} = \frac{{e^\infty}}{{\infty^2}} = \frac{\infty}{\infty}.$

Which is an indeterminate form ∞/∞.

Applying L'Hôpital's Rule:

First, we need to find the derivatives of the numerator and the denominator.

$\frac{{d}}{{dx}} e^x = e^x \quad \text{and} \quad \frac{{d}}{{dx}} x^2 = 2x.$

Now, we evaluate the limit of the quotient of the derivatives:

$\lim_{{x \to \infty}} \frac{{e^x}}{{2x}}.$

On differentiating we find that it is still in an indeterminate form ∞/∞, so apply

L'Hôpital's rule again:

$\frac{{d}}{{dx}} e^x = e^x \quad \text{and} \quad \frac{{d}}{{dx}} 2x = 2.$

Evaluate the limit of the quotient of the new derivatives:

$\lim_{{x \to \infty}} \frac{{e^x}}{2} = \infty.$

Conclusion:
$\lim_{{x \to \infty}} \frac{{e^x}}{{x^2}} = \infty.$

The limit is infinity.

Example 2:

Evaluate the following limit:
$\lim_{x \to 0} \frac{e^{x^2} - 1}{x^2}$

Direct Substitution:

If we substitute katx=0 into the expression, we get

$\frac{e^{0^2} - 1}{0^2} = \frac{1-1}{0} = \frac{0}{0}$

which is an indeterminate form 0/0. So, we can apply L'Hôpital's rule.

Applying L'Hôpital's Rule:

First, we need to find the derivatives of the numerator and the denominator.
$\frac{d}{dx} (e^{x^2} - 1) = 2x e^{x^2} \quad \text{and} \quad \frac{d}{dx} x^2 = 2x$

Now, we evaluate the limit of the quotient of the derivatives:

$\lim_{x \to 0} \frac{2x e^{x^2}}{2x} = \lim_{x \to 0} e^{x^2} = e^{0^2} = e^0 = 1$

So,

$\lim_{x \to 0} \frac{e^{x^2} - 1}{x^2} = 1$

## Applications of L’hopital’s rule:

Here are some common applications:

Analysis: Used in real and complex analysis to study the behavior of functions near singular points or asymptotes.

Physics: Calculating instantaneous rates of change in kinematics, such as finding velocity and acceleration from position-time graphs.

Engineering: Analyzing the behavior of circuits and systems in electrical engineering, particularly in the study of signals and systems.