# L’ Hospital’s rule

### L'Hôpital's Rule:

Definition: L'Hôpital's Rule is a powerful tool used to evaluate limits involving indeterminate forms $\frac{0}{0}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$. It enables the determination of certain limits by taking the derivatives of the numerator and denominator and re-evaluating the limit.

Indeterminate Forms L'Hôpital's Rule Applies To:

1. $\frac{0}{0}$ form: Occurs when both the numerator and denominator of a function approach zero as the variable approaches a certain value.
2. $\frac{\mathrm{\infty }}{\mathrm{\infty }}$ form: Arises when both the numerator and denominator of a function grow unbounded as the variable approaches a certain value.

L'Hôpital's Rule Statement: If $f\left(x\right)$ and $g\left(x\right)$ are differentiable functions on an open interval except possibly at a point $c$ (where $c$ can be a real number or $±\mathrm{\infty }$), and if ${\mathrm{lim}}_{x\to c}f\left(x\right)={\mathrm{lim}}_{x\to c}g\left(x\right)=0$, then:

${\mathrm{lim}}_{x\to c}\frac{f\left(x\right)}{g\left(x\right)}={\mathrm{lim}}_{x\to c}\frac{{f}^{\mathrm{\prime }}\left(x\right)}{{g}^{\mathrm{\prime }}\left(x\right)}$ If the limit on the right-hand side exists or is $±\mathrm{\infty }$.

Steps for Applying L'Hôpital's Rule:

1. Verify that the limit is in an indeterminate form of $\frac{0}{0}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$.
2. Take the derivatives of the numerator and denominator separately.
3. Re-evaluate the limit using the derivatives obtained.
4. Repeat the process if the result still yields an indeterminate form until a determinate value is reached or the limit is proven to be divergent.

Precautions:

• L'Hôpital's Rule can only be applied when the conditions for its use are met.
• It doesn’t always guarantee a solution; in some cases, repeated applications might not resolve the limit.

Example: Evaluate the limit ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}$.

Solution:

1. Initial Evaluation: Direct substitution yields ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}=\frac{\mathrm{sin}\left(0\right)}{0}$, which is of the form $\frac{0}{0}$, indicating an indeterminate form.

2. Apply L'Hôpital's Rule:

• Differentiate the numerator and denominator separately:
• Derivative of $\mathrm{sin}\left(x\right)$ is $\mathrm{cos}\left(x\right)$.
• Derivative of $x$ is $1$.
• Now, the limit becomes ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{cos}\left(x\right)}{1}$ after applying L'Hôpital's Rule.
3. Re-evaluate the Limit:

• Substituting $x=0$ into $\mathrm{cos}\left(x\right)$ gives $\mathrm{cos}\left(0\right)=1$.
• Therefore, ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}={\mathrm{lim}}_{x\to 0}\frac{\mathrm{cos}\left(x\right)}{1}=1$.

So, using L'Hôpital's Rule, the original indeterminate limit ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}$ evaluates to $1$.