It says that the limit when we divide one function by another is the same after we take the derivative of each function (with some special conditions shown later).

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In symbols we can write:

limx→cf(x)g(x) = limx→cf’(x)g’(x)

The limit as x approaches c of "f-of−x over g-of−x" equals thethelimit as x approaches c of "f-dash-of−x over g-dash-of−x"

All we did is showroom that little dash mark ’ on each function, which means to take the derivative.

At x=2 we would normally get:

22+2−622−4 = 00

Which is indeterminate, so we are stuck. Or are we?

Let"s try L"Hôpital!

Differentiate both top and bottom (see Derivative Rules):

limx→2x2+x−6x2−4 = limx→22x+1−02x−0

Now we just substitute x=2 khổng lồ get our answer:

limx→22x+1−02x−0 = 54

Here is the graph, notice the "hole" at x=2: Note: we can also get this answer by factoring, see Evaluating Limits.

Normally this is the result:

limx→∞exx2 =

Both head to infinity. Which is indeterminate.

But let"s differentiate both top và bottom (note that the derivative of ex is ex):

limx→∞exx2 = limx→∞ex2x

Hmmm, still not solved, both tending towards infinity. But we can use it again:

limx→∞exx2 = limx→∞ex2x = limx→∞ex2

Now we have:

limx→∞ex2 = ∞

It has shown us that ex grows much faster than x2.

## Cases

We have already seen a 00 & example. Here are all the indeterminate forms that L"Hopital"s Rule may be able lớn help with:

00 0×∞ 1∞ 00 ∞0 ∞−∞

## Conditions

### Differentiable

For a limit approaching c, the original functions must be differentiable either side of c, but not necessarily at c.

Likewise g’(x) is not equal lớn zero either side of c.

### The Limit Must Exist

This limit must exist:

limx→cf’(x)g’(x)

Why? Well a good example is functions that never settle to lớn a value.

Which is a case. Let"s differentiate top và bottom:

limx→∞1−sin(x)1

And because it just wiggles up and down it never approaches any value.

So that new limit does not exist!

And so L"Hôpital"s Rule is not usable in this case.

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BUT we can bởi this:

limx→∞x+cos(x)x = limx→∞(1 + cos(x)x)

As x goes khổng lồ infinity then cos(x)x tends to lớn between −1 & +1, và both tend to zero.