L’ hospital’s rule is as follows:
L’ hospital’s rule is used to find limits of rational functions where the indeterminate form () arises.
Example 1 – Find the limit of
If x increases without bound for the numerator and denominator, you would obtain:
This is the indeterminate form and is not a valid answer. Applying the l’ hospital’s rule:
f’(x) = 2x
g’(x) = ex
The indeterminate form still appears. The derivative must be applied again to the numerator and denominator:
f’(x) = 2
g’(x) = ex
Essentially, the denominator is increasing faster than the numerator, so the limit is getting closer to zero.
Example 2 – Find the limit of
If x increases without bound for the numerator and denominator, you would obtain:
This is the indeterminate form and is not a valid answer. Applying the l’ hospital’s rule:
The numerator is decreasing to zero. The denominator is increasing to infinity. The entire rational expression, therefore, is decreasing to zero as x increases without bound.
Example 3 – Find the limit of
If x increases without bound for the numerator and denominator, you would obtain . This is the indeterminate form and is not a valid answer. Applying the l’ hospital’s rule:
f’(x) = 7x6 + 9x2 + 16x
g’(x) = 2x + 4
The indeterminate form still appears. The derivative must be applied again to the numerator and denominator:
f’(x) =35x5+18x+16
g’(x) = 2
Essentially, the numerator is increasing faster than the denominator, so the entire expression will increase to infinity as x increases without bound.