Question

use l'hospital rule for this limits
1) lim(1+2/x)^3x approch x to infinity
please respond to me

Answer 1

L'HR cannot be used immediately. first, let \[y=\lim_{x \rightarrow +\infty} \left(1+\frac{2}{x}\right)^{3x}\]

Answer 2

okay then i will use ln for both side

Answer 3

right.

Answer 4

then i don't know to complete please could you solve it

Answer 5

i'll do some parts. \[\ln y=\ln \lim_{x \rightarrow +\infty}\left(1+\frac{2}{x}\right)^{3x}= \lim_{x \rightarrow +\infty}\ln \left(1+\frac{2}{x}\right)^{3x}\]

Answer 6

2\x

Answer 7

and the lim xrightarrow infty

Answer 8

apply the property of the \(\ln\) function,\[\ln y=\lim_{x \rightarrow +\infty} 3x \ln \left(1+\frac{2}{x}\right)\]and rewrite as \[\ln y=3\lim_{x \rightarrow +\infty} \frac {\ln \left(1+\frac{2}{x}\right)}{\frac{1}{x}}\]

Answer 9

now, it's your turn to apply the L'HR since the rational expression is an indeterminate form \[\left[ \frac{0}{0}\right]\]

Answer 10

okay thank you

Answer 11

actually, it will be easier use the define of