Question

Limits question

Answer 1

\[\Large \lim_{x \to \infty} \frac{f(x)}{g(x)}=1\] Does this imply \[\Large \lim_{x \to \infty} \frac{e^{f(x)}}{e^{g(x)}}=1\]

Answer 2

The answer is no, but I don't know how to show this.

Answer 3

draw a picture to clear it up and help you out :p

Answer 4

counter example...let f(x)=x and g(x)=x+1

Answer 5

\[\frac{x}{x+1}\to1\]

\[\frac{e^x}{e^{x+1}}=e^{-1}\neq 1\]

Answer 6

Awesome, thanks.

Answer 7

for a limit not involving infinity, e.g.
\[ \Large \lim_{x \to c} \frac{f(x)}{g(x)}=1 \]
we can say
\[ \lim_{x \to c} \frac{f(x)}{g(x)}=\frac{ \lim f(x)}{\lim g(x)} \]
but if the limit of f(x) and g(x) is infinity we get \( \frac{\infty}{\infty} \)
which is not defined