L'Hospital's Rule

0:00 / 0:00

L'Hospital's Rule
*used to handle indeterminate forms of a limit

Indeterminate Forms
 00\frac{0}{0}00​, ∞∞\frac{\infty}{\infty}∞∞​,  0×∞0\times\infty0×∞,  ∞−∞\infty-\infty∞−∞,  1∞1^{\infty}1∞,  ∞0\infty^0∞0,  000^000

L'Hopital's Rule 
Criteria:
fff and ggg are differentiable and g′(x)≠0g'\left(x\right)\ne0g′(x)=0 and 
lim⁡x→af(x)g(x)=00   or   ±∞±∞\lim_{x\to a}\frac{f\left(x\right)}{g\left(x\right)}=\frac{0}{0}\ \ \ or\ \ \ \frac{\pm\infty}{\pm\infty}limx→a​g(x)f(x)​=00​   or   ±∞±∞​
Then lim⁡x→a f(x)g(x)=lim⁡x→a f′(x)g′(x)\lim_{x\to a}\ \frac{f\left(x\right)}{g\left(x\right)}=\lim_{x\to a}\ \frac{f'\left(x\right)}{g'\left(x\right)}limx→a​ g(x)f(x)​=limx→a​ g′(x)f′(x)​.

Other Indeterminate forms

Practice: L'Hospital's Rule
Evaluate lim⁡x→∞ x3 sin⁡(1x2)\displaystyle \lim_{x\to\infty}\ x^3\ \sin\left(\frac{1}{x^2}\right)x→∞lim​ x3 sin(x21​).

000

1to1toratio,Orthatforeverytwosliceseverytwoslices1.96gpermole1 to 1 to ratio, Or that
for every two slices every two slices 1.96 g per mole1to1toratio,Orthatforeverytwosliceseverytwoslices1.96gpermole

−1-1−1

It does not exist (approaches ∞\infty∞)

It does not exist (does not approach a single value)

I don't know

Practice: L'Hospital's Rule
Evaluate lim⁡x→0+(e1x)1ln⁡x\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}limx→0+​(ex1​)lnx1​.

Wize University Calculus 2 Textbook > Improper Integrals

Popular Courses

L'Hospital's Rule

Indeterminate Forms

L'Hopital's Rule

Other Indeterminate forms

Practice: L'Hospital's Rule

Practice: L'Hospital's Rule

Practice: L'Hospital's Rule

Extra Practice

Practice: L'Hospital's Rule

L'Hospital's Rule: Indeterminate Forms