4.4 Indeterminate Forms and L'Hospital's Rule

Hey everyone. I guess I’m here to help you all learn about the wonders of section 4.4 – Indeterminate Forms and L’Hospital’s Rule.

The point of this lesson is to find out what’s going on near the ends of the graph at infinity.

Okay, I guess we’ll be doing an example problem to learn this lesson quickly.

Example Problem:

We’re going to solve this using L’Hospital’s Rule:

***L’Hospital’s Rule***

Suppose f and g are differentiable and

g’(x)≠ 0 near a (except possibly at a). Suppose that:

and

or that

and

(In other words, we have an indeterminate form of typeor

or

If the limit on the right side exists (or is infinite or negative infinite).

Now, you always have to remember that the limit at infinity depends on how fast the top grows compared to how fast the bottom grows. And another thing to keep in mind, L’Hospital’s Rule can only be used when the limits are indeterminate.

There are 7 indeterminate forms of limits that we know of:

or

or

Although there are 7 indeterminate forms of limits, it’s usually best to try to change any the last 5 forms to one of the first two forms :

Now…you may be asking, “HOW do we do this?” It’s simple. First of all, we can see that if we had left the equation (ln x) / (x-1) alone and tried to find the limit by simply plugging “infinity” into ‘x’, the limit would become infinitely large. Eventually, “ln x” would become “infinity,” and “x-1” would also become “infinity.” Hopefully you realize that is one of our indeterminate forms. Now we know that we can use L’Hospital’s Rule!

So using L’Hospital’s Rule, we rewrite the derivative of “ln x” as (1/x)(1) and the derivative of “x-1” as 1.

To find the derivative of this whole limit of a function, we don’t use the quotient rule. Instead, we take the derivative of the top (ln x) and divide it by the derivative of the bottom (x-1).

Eventually, we can find that the limit, as x approaches infinity, is zero.

To simplify the definition of L’Hospital’s Rule, here’s an equation to help you figure this thing out:

Well, I hope you understand all of this…but in case you don’t, we’ll try some different problems that use The Rule (sorry, I just don’t want to constantly type out L’Hospital’s Rule…which I just did…again. AHHHHHH!!!).

Example:

Using THE Rule, we find that the derivative of (3x² + 4) is 6x, and the derivative of (2x² + 7) is 4x. So we just put the top derivative over the bottom derivative, like this , cancel out the x’s, do some simplifying…and PRESTO! We find that the limit is: Another problem like this:

We used the Rule to find that the limit, as x approaches infinity, is 4! Huzzah!

Okay, last sample problem…this one’s a little different from the last two we’ve had.

Example:

Without using the Rule, we would find that the limit would be:

Now, don’t make this mistake. Although you might think that zero multiplied with negative infinity would equal zero, it doesn’t. Zero multiplied with negative infinity is actually an indeterminate form of limits. That means that we can use the RULE!

So we will re-do this whole thing using L’Hospital’s Rule:

First, let’s start off by showing that the limit becomes an indeterminate form of limits.

We can rewrite x so that it does not end up equaling zero.

X =

Now that this is in an indeterminate form that we like, we use L’Hospital’s Rule:

Derivative of (ln x) divided by the derivative of (x^-1) So we see that as x approaches 0 from the right side, the limit becomes -(0).

And no, Jean, this does not “become negative zero”…it is simply zero.

Hopefully you all understand this lesson now because I know I DO!

BUT, in case you still need help, you can go to these sites:

Site#1

or

Site#2

Even though I had such a fun time using Equation Editor 41 times and uploading 35 pictures multiple times, it is definitely time to stop before I get addicted to blogging. So here’s a reminder to Jean that she is up next! Hurrah!


sorry about the blacked out equation...you can just click on it to see the equation...o, and sorry about this blue-underlined sentence...I don't know why it's doing this, but it's really annoying and I want it to stop...but it won't. So....I can't do anything.