# Thread: When to use L'Hospital's Rule...

1. ## When to use L'Hospital's Rule...

I'm having some issues determining when I can and cannot use L'Hospital's Rule. Tips? Suggested approaches?

2. You can use it when you have the following typical cases:

#1 Indetermination of the form $\frac00.$

#2 Indetermination of the form $\frac\infty\infty.$

Of course, if you're solving a problem and this one says that you cannot apply the Rule, don't do it

Many limits can be computed without the Rule, that makes them more interesting.

3. Just to add to his comment, there are sometime that you manipulate the problem to get to 0/0 or ∞/∞
Such cases are when the limits become one of those forms : 0^0 1^∞ ∞ - ∞, 0⋅∞ ∞^0

4. That's what I'm having issues with -- problems where it isn't apparent that l'hospital's rule applies. My professor just told me to practice, but I'm not getting anywhere...I feel like I'm missing something, (probably extremely simple), but we glossed over the material so fast, I'm not surprised I'm having a little difficulty.

5. when you plug x in and you get any of the form I listed above, usually there is a way to manipluated it to 0/0 or $\frac\infty\infty
$

then you can use the L'Hospital's rule

6. For example, the limit as t approaches 0 of (e^t)-1/t^3

It looks like "0/0" so I use l'hopital's rule and get (e^t)/3t^2

Looking at my calculator, I know it's infinity, but I don't understand how we get that from using the rule. I get 1/0 then...

7. People sometimes apply the Rule thousand of times, but always you need to check if you got indetermination.

So, when you evaluate directly $\frac{e^t}{3t^2},$ there's no indetermination, and the conclusion follows.

8. Originally Posted by Esiuol
For example, the limit as t approaches 0 of (e^t)-1/t^3

It looks like "0/0" so I use l'hopital's rule and get (e^t)/3t^2

Looking at my calculator, I know it's infinity, but I don't understand how we get that from using the rule. I get 1/0 then...
hmm you probably forgot this...
Lim as t-->0 for (e^t)/3t^2 = infinity because at as the denominator goes to zero, the number gets bigger and bigger...
So yea, the L'hospital rule did work for that problem

9. Can I also use L'Hospital's rule in:
a) $\frac{-\infty}{\infty}$

b) $\frac{\infty}{-\infty}$

c) $\frac{-\infty}{-\infty}$

d) $\frac{-0}{0}$

e) $\frac{0}{-0}$

f) $\frac{-0}{-0}$

(added the zeros, figure they all take the form 0/0, but may as well make sure )

10. Can I also use L'Hospital's rule in:
a)

b)

c)

d)

e)

f)
Yes because:

Lim (-f(x)) = -Lim(f(x))
and $
\frac {d}{dx} (-f(x))$
= $-\frac{d}{dx}(f(x))$

Such cases are when the limits become one of those forms : 0^0 1^∞ ∞ - ∞, 0⋅∞ ∞^0
each of these cases have general methods which you would do well to memorise:
$
0^0$
:
$lim 0^0 = e^{lim log(0^0)}$
= $e^{lim 0log0}$
then use the method for 0⋅∞

$1^\infty$:
lim $1^\infty$ = $e^{lim log(1^{\infty})}$
= $e^{lim \infty log1}$
then use the method for 0⋅∞

$\infty^0$:
$lim \infty^0$ = $e^{lim log (\infty^0)}$
= $e^{lim 0log(\infty)}$
then use method for 0⋅∞

0⋅∞:
Do either of the following:
0⋅∞ = $\frac{0}{1/\infty}$
=0/0
or
0⋅∞ = $\frac {\infty}{1/0}$
= $\infty/\infty$

$\infty - \infty$:
$lim \infty-\infty = log(lim e^{\infty-\infty})$ $
=log (lim \frac{e^\infty}{e^\infty})$

$=log (lim \infty/\infty)$

11. Originally Posted by badgerigar
Yes because:

Lim (-f(x)) = -Lim(f(x))
and $
\frac {d}{dx} (-f(x))$
= $-\frac{d}{dx}(f(x))$

each of these cases have general methods which you would do well to memorise:
$
0^0$
:
$lim 0^0 = e^{lim log(0^0)}$
= $e^{lim 0log0}$
then use the method for 0⋅∞

$1^\infty$:
lim $1^\infty$ = $e^{lim log(1^{\infty})}$
= $e^{lim \infty log1}$
then use the method for 0⋅∞

$\infty^0$:
$lim \infty^0$ = $e^{lim log (\infty^0)}$
= $e^{lim 0log(\infty)}$
then use method for 0⋅∞

0⋅∞:
Do either of the following:
0⋅∞ = $\frac{0}{1/\infty}$
=0/0
or
0⋅∞ = $\frac {\infty}{1/0}$
= $\infty/\infty$

$\infty - \infty$:
$lim \infty-\infty = log(lim e^{\infty-\infty})$ $
=log (lim \frac{e^\infty}{e^\infty})$

$=log (lim \infty/\infty)$
I see, my book was so vague about what cases I could use it, I remember several times that I got it into usable form, but I didn't think it was usable and kept manipulating it until I got something crazy that would work. And when I finished I would look at it bewildered and go "I really hope that's not on the test"