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| === Other Cases === | | === Other Cases === |
− | See http://calculus.seas.upenn.edu/?n=Main.LHopitalsRule | + | * See http://calculus.seas.upenn.edu/?n=Main.LHopitalsRule |
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| + | == Links == |
| + | * http://math.stackexchange.com/questions/584889/the-intuition-behind-lhopitals-rule/ |
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| == 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 Big-O notation
Other Cases
Links
Sources