**1.** Now we are concerned about limits and we can have nightmarish limits. For example if

(1a)

then

(1b)?

What can we do?

We end up using derivatives to help evaluate such limits. We have L’Hopital’s rule state, if , then

(1c)

**2. Sketch of Proof.** We can use linear approximations writing

(2) and

Use these approximations in (1a), then take the limit . So our limit becomes

(3)

We evaluate the inner limit first

(4)

But look , so (4) simplifies to

(5)

Thus we obtain

(6)

So we obtain

(7)

**3. Sketch of Proof for .** When

(8a)

and

(8b)

we run into similar problems. L’Hopital’s rule still holds. We just do the following trick:

(9)

and we recover the previous case…kind of!

Taking the derivative of top and bottom, we get

(10)

We can cross multiply, giving us

(11)

Cross multiplying gives us

(12)

and thus

(13).

**Notation:** We will write

(14)

when we apply L’Hopital’s rule.

**Example 1.** Consider the sequence

(15).

What happens as ? We have to apply L’Hopital’s rule *twice*

(16).

Thus we have

(17).

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