How to take the derivative of this function

[JKLM]

What is the Lim as x approaches 0 of (1+sin(pi*x))^(1/x)
I couldn't figure out how to take the derivative of this function

 

[mathman]

You don't need to take derivatives.

As x->0, sin(pi*x) is approx pi*x. Your expression can be represented as exp(ln(1+pi*x)/x). Then ln(1+pi*x) is approx pi*x as x->0. The net result is exp(pi) is the limit that you want.

 

[himanshu121]

for the derivative part take log
say
$$L = \lim_{x\rightarrow 0}[1+sin(\pi x)]^\frac{1}{x}$$

Take log
$$logL = \lim_{x\rightarrow 0}\frac{log(1+sin\pi x)}{x}$$
I hope u can differentiate from there on

Though mathman's reply is best u should do it that way