A rational function is the ratio of two functions, f(x) and g(x).

To find lim_x-->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.

L'Hopital's Rule

If f(x) and g(x) are differentiable on the interval (a,b) which contains c, except possibly at c itself, and lim_x->cf(c) = lim_x->cg(c) = 0, then:

lim_x-->c (f(x)/g(x)) = lim_x-->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.

Example

Evaluate lim_x-->0 sin(2x)/5x

Since the lim_x-->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:

2cos(2x)/5

As x->0, this approaches 2/5.

Internet references

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

Related pages in this website

Limits

Definition of Continuous

The webmaster and author of the Math Help site is Graeme McRae.
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