In order that the function `f(x)=(x+1)^(cotx)` is continuous at x = 0, f(0) must be defined as

Options

A. `f(0)=(1)/(e)`
B. `f(0)=0`
C. `f(0)=e`
D. None of these

Answer

C

Explanation

Since, the fuction at x = 0 is continuous, we must have
`f(0)=underset(xto0)limf(x)`
`=underset(xto0)lim(x+1)^(cotx)=underset(xto0)lim{(1+x)^((1)/(x))}^(xcotx)`
`=underset(xto0)lim{(1+x)^((1)/(x))}^(underset(xto0)lim((x)/(tanx)))=e^(1)=e`
`[becauseunderset(xto0)lim(1+x)^(1//x)=e" and "underset(xto0)lim(x)/(tanx)=1]`