

Line 43: 
Line 43: 
   
 === Other Cases ===   === Other Cases === 
−  See http://calculus.seas.upenn.edu/?n=Main.LHopitalsRule  +  * See http://calculus.seas.upenn.edu/?n=Main.LHopitalsRule 
   
 +  == Links == 
 +  * http://math.stackexchange.com/questions/584889/theintuitionbehindlhopitalsrule/ 
   
 == Sources ==   == Sources == 
Latest revision as of 14:05, 2 January 2016
L'Hôpital's Rule
The rule has this form:
$$\lim_{x \to a} \cfrac{f(x)}{g(x)} = \lim_{x \to a} \cfrac{f'(x)}{g'(x)}$$
$0/0$ case
 suppose $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \cfrac{0}{0}$
 i.e. $f(a) = 0$ and $g(a) = 0$
 if $f(x)$ and $g(x)$ are continuous
 then the rule is $$\lim_{x \to a} \cfrac{f(x)}{g(x)} = \lim_{x \to a} \cfrac{f'(x)}{g'(x)}$$
Can show this using Taylor Expansion about $x = a$:
 $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f(a) + f'(a)\, (x  a) + \ ...}{g(a) + g'(a)\, (x  a) + \ ...}$
 know that $f(a) = g(a) = 0$, so have
 $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a)\, (x  a) + \ ...}{g'(a)\, (x  a) + \ ...}$
 can factor $(x  a)$ out, so now we have:
 $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a) + \ ...}{g'(a) + \ ...}$
 the leading order terms are $f'(a)$ and $g'(a)$, and the rest vanish under the limit
Examples:
 $\lim\limits_{x \to 0} \cfrac{\sin x}{x} = \lim\limits_{x \to 0} \cfrac{\cos x}{1} = 1$
 $\lim\limits_{x \to 0} \cfrac{1  \cos x}{x} = \lim\limits_{x \to 0} \cfrac{\sin x}{1} = 0$
$\infty / \infty$ case
 suppose $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \cfrac{\infty}{\infty}$
 i.e. $f(a) = \infty$ and $g(a) = \infty$
 if $f(x)$ and $g(x)$ are continuous
 then the rule is $$\lim_{x \to a} \cfrac{f(x)}{g(x)} = \lim_{x \to a} \cfrac{f'(x)}{g'(x)}$$
Example:
 $\lim\limits_{x \to \infty} \cfrac{\ln x}{\sqrt{x}}$
 both go to $\infty$, but the rate at which they approach $\infty$ is different
 by taking the derivative, we can see which one grows faster
 is this case, $\sqrt{x}$ dominates $\ln x$: it grows much faster
 so the limit is 0
To say that one function grows faster than other, we can use the BigO notation
Other Cases
Links
Sources