Question

Find the Limit. Use l'hospital's rule where appropriate.

lim as x goes to 0 from the right of (ln x / x)

I did l'hospital and got...
lim as x goes to 0 from the right of (1/x)/1, but that is still indeterminate

so i did it again, but the denominator would = 0 so we can't divide by 0

the book answer is - infinity, but i dont understand why

Answer 1

0 = cos(x) -sin(x)
cos(x) = sin(x)
x = 45, 225

Answer 2

One thing you might want to keep in mind, even though it will not speed this problem on its way. \[\sin(x) + \cos(x) = \sqrt{2}\cdot\cos\left(x-\frac{\pi}{4}\right)\] Just keep it somewhere in the back of your head. You may need the general form, some day.

Answer 3

All right, so:
\[
\lim_{x\to0^+}\left(\frac{\ln(x)}{x}\right)=\lim_{x\to0^+}\left(\frac{\frac{d}{dx}\ln(x)}{\frac{d}{dx}x}\right)=\lim_{x\to0^+}\left(\frac{\frac{1}{x}}{1}\right)=\lim_{x\to0^+}\left(\frac{1}{x}\right)=\infty
\]Are you sure it's not from the left?

Answer 4

Yes, I'm sure. So... in your answer... since x is not ACTUALLY 0 (just very close) we get infinity because it's 1/(a very very small number)?