Question

find the limit as x approaches 0 of (cosx+2x)^1/x

Answer 1

first recognize that e^ln(x) = x
so
\[\large e^{\lim_{x \rightarrow 0} \ln(\cos x +2x)^{1/x}}\]
\[\rightarrow \large e^{\lim_{x \rightarrow 0} \frac{\ln(\cos x +2x)}{x}}\]
this yields lim 0/0 which is indeterminate
use L'Hopitals rule, take derivative of top/bottom
\[\rightarrow \large e^{\lim_{x \rightarrow 0} \frac{2-\sin x}{\\cos x +2x}}\]
\[= e^2\]

Answer 2

isn't the derivative of ln(cosx+2x) 1/(cosx+2x) *-sinx +1

Answer 3

no derivative of 2x is 2

Answer 4

oh I forgot to take that derivative properly..

Answer 5

Thank you so much, I was having trouble finding out what I did wrong.

Answer 6

no problem