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) = 7x^{6} + 9x^{2} + 16x

g’(x) = 2x + 4

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

f’(x) =35x^{5}+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.