## Other Indeterminate Quotient Forms

We have discussed the indeterminate quotient form . But there some indeterminate quotient forms similar to
this one. Indeed, since

then one can see that the indeterminate quotient forms

as well as the indeterminate forms

may be easily converted to the indeterminate quotient form . So first notice that Hôpital's rule is still valid
when dealing with the indeterminate quotient form . Also worth to mention that the point
*a* may be finite or infinite, Hôpital's rule still applies.

**Example.** Find the limit

**Answer.** Fix . We have

Hence

We will use Hôpital's rule. We have

So clearly we will keep to use Hôpital's rule n-times to get to the function

Since

Therefore, we have

Note that this limit implies that, though both functions are very
large when , the exponential function is more
powerful than the power function (in fact more powerful than any
polynomial function). We will write

**Example.** Find the limit

**Answer.** Fix . We have

Hence

We will use Hôpital's rule. We have

Hence we have

which implies

Clearly this example implies that

Putting the two examples together we conclude that

when .

**The indeterminate forms **

The main idea behind these indeterminate forms is to transform 0 into
(the depends on whether we have 0+ or
0-), or transform into .
This will lead to the indeterminate quotient forms

Practically, you will be given a product *f*(*x*)*g*(*x*) where one function
goes to 0 while the other one goes to . So you will use
the following algebraic manipulations

**Example.** Find the limit

**Answer.** We have

Hence

Rewrite the given expression into

Computing the limit we will find

Here we have a choice. We may use Hôpital's rule or Taylor Polynomials. In any case, Hôpital's rule is not bad to use in this
case. Indeed, we have

Since

we conclude that

Note that when , we have

and

which imply

May be this is easier, what do you think???

**Example.** Find the limit

**Answer.** We have again

We will ask you to check that whether you take

or

The calculations are not easy. Here let us show how some tricks may
help. First switch from *x* into *t* = 1/*x*. We will have

Note that when . Next, we
use

(when ), to get

We already proved that

Therefore, we have

**
**

**
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