|
|
|
|
l'Hôpital's rule
|
(Theorem)
|
|
|
L'Hôpital's rule states that given an unresolvable limit of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ the ratio of functions $\frac{f(x)}{g(x)}$ will have the same limit at $c$ as the ratio $\frac{f'(x)}{g'(x)}$ In short, if the limit of a ratio of functions approaches an indeterminate form, then $$\lim_{x\rightarrow c}\frac{f(x)}{g(x)} = \lim_{x\rightarrow c}\frac{f'(x)}{g'(x)}$$ provided this last limit exists. L'Hôpital's rule may be
applied indefinitely as long as the conditions are satisfied. However it is important to note, that the nonexistance of $\lim\frac{f'(x)}{g'(x)}$ does not prove the nonexistance of $\lim\frac{f(x)}{g(x)}$
Example: We try to determine the value of $$\lim_{x\to \infty}\frac{x^2}{e^x}.$$ As $x$ approaches $\infty$ the expression becomes an indeterminate form $\frac{\infty}{\infty}$ By applying L'Hôpital's rule twice we get $$\lim_{x\to\infty}\frac{x^2}{e^x}=\lim_{x\to \infty}\frac{2x}{e^x}=\lim_{x\to \infty}\frac{2}{e^x}=0.$$ Another example of the usage of L'Hôpital's rule can be found here.
|
"l'Hôpital's rule" is owned by mathwizard. [ full author list (2) | owner history (1) ]
|
|
(view preamble | get metadata)
Cross-references: expression, indeterminate form, functions, ratio, limit
There are 11 references to this entry.
This is version 10 of l'Hôpital's rule, born on 2002-02-25, modified 2004-04-07.
Object id is 2657, canonical name is LHpitalsRule.
Accessed 31063 times total.
Classification:
| AMS MSC: | 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems) | | | 26C15 (Real functions :: Polynomials, rational functions :: Rational functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|