(Created page with "== 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...") |
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** know that $f(a) = g(a) = 0$, so have | ** know that $f(a) = g(a) = 0$, so have | ||
* $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a)\, (x - a) + \ ...}{g'(a)\, (x - a) + \ ...}$ | * $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a)\, (x - a) + \ ...}{g'(a)\, (x - a) + \ ...}$ | ||
− | * can factor $(x - a)$ out, so we have: | + | * can factor $(x - a)$ out, so now we have: |
* $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a) + \ ...}{g'(a) + \ ...}$ | * $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a) + \ ...}{g'(a) + \ ...}$ | ||
* the leading order terms are $f'(a)$ and $g'(a)$, and the rest vanish under the limit | * the leading order terms are $f'(a)$ and $g'(a)$, and the rest vanish under the limit | ||
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* so the limit is 0 | * so the limit is 0 | ||
− | To say that one function grows faster than other, we can use the [[ | + | To say that one function grows faster than other, we can use the [[Orders of Growth|Big-O notation]] |
=== 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/the-intuition-behind-lhopitals-rule/ | ||
== Sources == | == Sources == |
The rule has this form: $$\lim_{x \to a} \cfrac{f(x)}{g(x)} = \lim_{x \to a} \cfrac{f'(x)}{g'(x)}$$
Can show this using Taylor Expansion about $x = a$:
Examples:
Example:
To say that one function grows faster than other, we can use the Big-O notation