# 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:

- lim x→a f(x) = 0 or lim x→a f(x) = ∞
- lim x→a g(x)=0 or lim x→a g(x) = ∞
- 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:

- Verify that the limit is in the form
**0/0**or**∞/∞**. - Differentiate the numerator and denominator separately.
- Calculate the limit of the quotient of the derivatives.
- 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.

## References:

Khan Academy. (n.d.). L’Hoptial’s Rule introduction. Khan Academy.

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