I'm having some issues determining when I can and cannot use L'Hospital's Rule. Tips? Suggested approaches?
You can use it when you have the following typical cases:
#1 Indetermination of the form $\displaystyle \frac00.$
#2 Indetermination of the form $\displaystyle \frac\infty\infty.$
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.
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.
Can I also use L'Hospital's rule in:
a) $\displaystyle \frac{-\infty}{\infty}$
b) $\displaystyle \frac{\infty}{-\infty}$
c) $\displaystyle \frac{-\infty}{-\infty}$
d) $\displaystyle \frac{-0}{0}$
e) $\displaystyle \frac{0}{-0}$
f) $\displaystyle \frac{-0}{-0}$
(added the zeros, figure they all take the form 0/0, but may as well make sure )
Yes because:
Lim (-f(x)) = -Lim(f(x))
and$\displaystyle
\frac {d}{dx} (-f(x))$ = $\displaystyle -\frac{d}{dx}(f(x))$
each of these cases have general methods which you would do well to memorise:Such cases are when the limits become one of those forms : 0^0 1^∞ ∞ - ∞, 0⋅∞ ∞^0
$\displaystyle
0^0$:
$\displaystyle lim 0^0 = e^{lim log(0^0)}$
=$\displaystyle e^{lim 0log0}$
then use the method for 0⋅∞
$\displaystyle 1^\infty$:
lim $\displaystyle 1^\infty$ =$\displaystyle e^{lim log(1^{\infty})}$
= $\displaystyle e^{lim \infty log1}$
then use the method for 0⋅∞
$\displaystyle \infty^0$:
$\displaystyle lim \infty^0$ = $\displaystyle e^{lim log (\infty^0)}$
=$\displaystyle e^{lim 0log(\infty)}$
then use method for 0⋅∞
0⋅∞:
Do either of the following:
0⋅∞ = $\displaystyle \frac{0}{1/\infty}$
=0/0
or
0⋅∞ = $\displaystyle \frac {\infty}{1/0}$
=$\displaystyle \infty/\infty$
$\displaystyle \infty - \infty$:
$\displaystyle lim \infty-\infty = log(lim e^{\infty-\infty})$$\displaystyle
=log (lim \frac{e^\infty}{e^\infty})$
$\displaystyle =log (lim \infty/\infty)$
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"