A rational function is the ratio of two functions, f(x) and g(x).
To find limx−>c f(x)/g(x), where f(x) and g(x) are continuous
functions at x=c, then first look at
f(c)/g(c). If g(c) is not zero, then the limit is simply f(c)/g(c).
If g(c) is zero, then if f(c) is not zero, the limit is undefined
Finally, if both f(c) and g(c) are zero, then the rational function has the indeterminate form of 0/0, and L'Hopital's Rule can be used to find the limit of the rational function.
If f(x) and g(x) are differentiable on the interval (a,b) which contains c, except possibly at c itself, and limx−>cf(c) = limx−>cg(c) = 0, then:
limx−>c (f(x)/g(x)) = limx−>c (f'(x)/g'(x)),
as long as f'(x) and g'(x) don't change sign infinitely often in a neighborhood of c.
From this version of the rule, it's possible to prove other variants of it, for one-sided limits, limits as x approaches plus or minus infinity, or when the limits of f and g are both infinite. In all cases,
lim (f(x)/g(x)) = lim (f'(x)/g'(x))
See the Mathworld article on L'Hospital's Rule (an alternative spelling) for more information.
Evaluate limx−>0 sin(2x)/5x
Since the limx−>0 sin(2x)/5x gives the indeterminate form, 0/0, you have to take the derivative of the numerator, sin(2x), and the denominator, 5x, and then divide:
d/dx sin(2x) = 2cos(2x)
d/dx 5x = 5
Set up the quotient:
As x−>0, this approaches 2/5.
Karl's Calculus Tutor: L'Hopital's Rule (and the Cheshire Cat's Grin)
Analyze Math: Limits, Indeterminate Form and L'Hopital's Rule
Mathworld: L'Hospital's Rule
Definition of Continuous
The webmaster and author of this Math Help site is Graeme McRae.