L’Hospital’s Rule
L’Hospital’s Rule
Suppose $f$ and $g$ are differentiable and $g^{\prime}(x) \neq 0$ near $x=c$ (except possibly at $c$ ).
If the limit of $\frac{f(x)}{g(x)}$ as $x$ approaches $c$ produces the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then
$
\lim _{x \rightarrow c} \frac{f(x)}{g(x)}=\lim _{x \rightarrow c} \frac{f^{\prime}(x)}{g^{\prime}(x)}
$
provided the limit on the right side exists.
L’Hospital’s Rule can be applied only to quotients leading to indeterminate forms such as
$\frac{0}{0}, \frac{\infty}{\infty}, 0 \cdot \infty, 1^{\infty}, \infty^0, 0^0$, and $\infty-\infty$.
L’Hospital’s Rule does not apply when either the numerator or denominator has a finite nonzero limit.
Example 1
- Find $\lim _{x \rightarrow 0} \frac{e^x-1}{\sin x}$
▶️Answer/Explanation
Solution
Indeterminate form $\frac{0}{0}$
L’Hospital’s Rule: $\frac{d}{d x}\left(e^x-1\right)=e^x, \frac{d}{d x}(\sin x)=\cos x$. $e^0=1$ and $\cos 0=1$
$e^0=1$ and $\cos 0=1$.
$\begin{aligned} & \lim _{x \rightarrow 0} \frac{e^x-1}{\sin x} \\ & =\lim _{x \rightarrow 0} \frac{e^x}{\cos x} \\ & =1\end{aligned}$
Example 2
- Find $\lim _{x \rightarrow \pi / 2} \frac{\sec x+9}{\tan x}$.
▶️Answer/Explanation
Solution
$\lim _{x \rightarrow \pi / 2} \frac{\sec x+9}{\tan x}$ Indeterminate form $\frac{\infty}{\infty}$.
$=\lim _{x \rightarrow \pi / 2} \frac{\sec x \tan x}{\sec ^2 x}$ L’Hospital’s Rule: $\frac{d}{d x}(\sec x+9)=\sec x \tan x$,
$\frac{d}{d x}(\tan x)=\sec ^2 x$
$=\lim _{x \rightarrow \pi / 2} \frac{\tan x}{\sec x}$ Simplify
$\begin{aligned} & =\lim _{x \rightarrow \pi / 2} \sin x \\ & =1\end{aligned}$
Example 3
- Find $\lim _{x \rightarrow \infty} x \tan \frac{1}{x}$.
▶️Answer/Explanation
Solution
$
\begin{array}{ll}
\lim _{x \rightarrow 0} \frac{e^x-1}{\sin x} & \text { Indeterminate form } \frac{0}{0} \\
=\lim _{x \rightarrow 0} \frac{e^x}{\cos x} & \text { L’Hospital’s Rule: } \frac{d}{d x}\left(e^x-1\right)=e^x, \frac{d}{d x}(\sin x)=\cos x . \\
=1 & e^0=1 \text { and } \cos 0=1 .
\end{array}
$
Example4
- $
\lim _{x \rightarrow 0} \frac{e^x-1-x}{x^2}=
$
(A) 0
(B) $\frac{1}{2}$
(C) 1
(D) $\infty$
▶️Answer/Explanation
Ans:B
Example5
- $
\lim _{x \rightarrow 0} \frac{\sin ^{-1} x}{x}=
$
(A) $-\infty$
(B) 0
(C) $\frac{\pi}{2}$
(D) 1
▶️Answer/Explanation
Ans:D
Example 6
$
\lim _{\theta \rightarrow \pi} \frac{\sin \theta}{\theta-\pi}=
$
(A) -1
(B) $-\frac{1}{2}$
(C) 0
(D) $\frac{1}{2}$
▶️Answer/Explanation
Ans:A