L’ Hospital’s Rule

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) = e^x

The indeterminate form still appears. The derivative must be applied again to the numerator and denominator:

f’(x) = 2
g’(x) = e^x

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.