Question

Value of limx → 0⁡(1+cot(x))^sin(x)

(a) e

(b) e^2

(c) ^1⁄e

(d) Can not be solved

The question was asked in a job interview.

I'm obligated to ask this question of Indeterminate Forms in division Differential Calculus of Engineering Mathematics

Answer 1

Right option is (a) e

Best explanation: \(\Rightarrow \lim_{x\rightarrow 0}(1+cot(x))^{sin(x)}=\lim_{x\rightarrow 0}(1+\frac{cos(x)}{sin(x)})^{sin(x)}\)

\(=\lim_{x\rightarrow 0}(1+\frac{cos(x)}{sin(x)})^{\frac{sin(x)}{cos(x)}cos(x)}\)

\(\Rightarrow \lim_{x\rightarrow 0}\left [(1+\frac{cos(x)}{sin(x)})^{\frac{sin(x}{cos(x)}}\right ]^{cos(x)} \)

\(\Rightarrow\) Put cos(x)/sin(x)=t gives

\(\Rightarrow \lim_{t\rightarrow 0}\left [(1+t)^{\frac{1}{t}} \right ] ^{\lim_{x\rightarrow 0}cos(x)}\)

=>e^1

=>e