L’Hopital’s Rule

What are Indeterminate Forms?

Applying L’Hopital’s Rule:

• Check that the limit of f(x) / g(x) is an indeterminate form 0 / 0 .

• Differentiate f and g separately.

• Find the limit of f’(x) / g’(x).

Indeterminate Form of Type 0 / 0Suppose that f and g are differentiable functions on an open interval containing x = a, except possibly at x = a, and that

How about some examples?Find the following limits:

1.

2.

3.

4.

Indeterminate Form of TypeSuppose that f and g are differentiable functions on an open interval containing x = a, except possibly at x = a, and that

How about some examples?Find the following limits:

1.

2.

3.

Indeterminate Form of TypeCan sometimes be evaluated by rewriting the product as a ratio:

1.

2.

Indeterminate Form of TypeCan sometimes be evaluated by combining the terms and manipulating the result to produce quotient

1.

2.

Indeterminate Form of TypeCan sometimes be evaluated by first introducing a dependent variable

And then computing the limit of lny. Since

The limit of lny will be an indeterminate form of type

How about an example?Show that

Let’s practice