Question

Find the limit (medal given for showing steps!):
\[\huge \lim_{x \rightarrow 1} \frac{5x}{x-1} - \frac{5}{ln x} \]
(I keep getting -∞ even after the derivation)

Answer 1

\[ \infty - \infty \text{ case, i can be changed into } 0/0 \text{ case if you manipulate it carefully. }\]

Answer 2

*it

Answer 3

\[ \huge \frac{5x}{x-1} - \frac{5}{\ln x} = \frac{5x \ln x - 5(x-1)}{(x-1)\ln x} \]
Now use L'hopital's rule

Answer 4

Yes, Agentx, I think this is a case of not being careful enough in applying L'Hopital's rule. To use L'H's, it is required that our limit be of one of these indeterminate forms:
\[\large \frac{0}{0}, \frac{\infty}{\infty}, \frac{-\infty}{-\infty}\]
With your problem, if you evaluate at 1, you get
\[\large \frac{5}{0}-\frac{5}{0}\] And you can't use L'Hopital's on either of those forms.

Answer 5

However, as experiment shows, you can manipulate these a bit so that they are in indeterminate form. =D

Answer 6

Thanks to you both! :D

Answer 7

My pleasure! =D

Answer 8

yw