Calculus I - L'Hôpital's rule and Indeterminate Forms
4 flashcards · Mathematics · by luckydog789 (harrison holbrook)
L'Hôpital's rule
Suppose the functions fff and ggg are differentiable on an open interval III containing the number ccc , except possibly at ccc , and g′(x)≠0g'(x) \neq 0g′(x)=0 for all x≠cx \neq cx=c in III . Let LLL denote either a real number of ±∞\pm \infty±∞ , and suppose f(x)g(x)\frac{f(x)}{g(x)}g(x)f(x) is an indeterminate form at ccc of the type 00\frac{0}{0}00 or ∞∞\frac{\infty}{\infty}∞∞ . If limx→cf′(x)g′(x)=L\lim_{x \to c} \frac{f'(x)}{g'(x)}=Llimx→cg′(x)f′(x)=L , then limx→cf(x)g(x)=L\lim_…
Steps for finding a limit using L'Hôpital's rule
Check that f(x)g(x)\frac{f(x)}{g(x)}g(x)f(x) is indeterminate at ccc of the form 00\frac{0}{0}00 or ∞∞\frac{\infty}{\infty}∞∞ Differentiate fff and ggg respectively Find limx→cf′(x)g′(x)\lim_{x \to c} \frac{f'(x)}{g'(x)}limx→cg′(x)f′(x) . This is equal to limx→cf(x)g(x)\lim_{x \to c} \frac{f(x)}{g(x)}limx→cg(x)f(x) , provided the limit exists or is ±∞\pm \infty±∞ if f′g′\frac{f'}{g'}g′f′ is indeterminate in the form of 00\frac{0}{0}00 or ∞∞\frac{\infty}{\infty}∞∞ at ccc , repeat the…
Indeterminate Forms
0⋅∞0 \cdot \infty0⋅∞ , ∞−∞\infty - \infty∞−∞ , 000^000 , 1∞1^{\infty}1∞ , ∞0\infty^0∞0
L'Hôpital's rule on other indeterminate forms
To perform L'Hôpital's rule on other indeterminate forms, algebraically/trigonometrically simplify them into either 00\frac{0}{0}00 or ∞∞\frac{\infty}{\infty}∞∞