Question

lim as x tends to infinity of (1+3/x)^x

Answer 1

You may wish to introduce a logarithm.

Answer 2

Recall the definition of the important limit:
\[ \lim_{x\rightarrow \infty} \left(1+\frac{a}{x} \right)^x=e^a\]

Answer 3

Oh yes maybe it is too easy to use that.. lol back to first principles I guess

Answer 4

Yes the logarithm trick is probably one of the easier ways to prove that limit. What you can do is let \[y=\lim_{x\rightarrow\infty}\left( 1+\frac{3}{x}\right)^x\]
then take the log of both sides :
\[\begin{aligned} \log y &= \log \left[ \lim_{x\rightarrow\infty}\left( 1+\frac{3}{x}\right)^x\right]\\
&=\lim_{x\rightarrow\infty}\log\left[\left( 1+\frac{3}{x}\right)^x\right]\\&
= \lim_{x\rightarrow\infty}x\log\left( 1+\frac{3}{x}\right)\\
& =\lim_{x\rightarrow\infty}\frac{\log\left( 1+\frac{3}{x}\right)}{\frac{1}{x}}\end{aligned}\]

This should give you a good start