Why is the limit as you approach infinity of (1+(3/x))^x equal to e^3 and not 1? It looks like as you approach infinity 3/x becomes 0 and 1 to the infinite power is just 1. I know how to do l'hopital's rule, that's not the issue here, I just don't understand why I have to go that route.

Question

kainui · Answer

$$\lim_{x \rightarrow \infty}[1+(3/x)]^x$$ to be clear, this is the limit.

anonymous · Answer

The answer is e^3

kainui · Answer

I know the answer, that's not what I'm looking for.

anonymous · Answer

$$
y=\left(\frac{3}{x}+1\right)^x\
\ln y  = x \ln \left(\frac{3}{x}+1\right) \
=\frac {\ln \left(\frac{3}{x}+1\right)}
{\frac 1 x}
$$
and apply L'Hospital rule to find that the limit of ln (y) is 3 and conclude.

kainui · Answer

That didn't answer my question at all.

anonymous · Answer

$$ 1^\infty 
$$
is undetermined. It is not always  equal to 1.

kainui · Answer

Thanks, that's all I needed.

anonymous · Answer

With pleasure. I should have read your whole question. It would have saved ne some typing.