Question

If f(x) = e^x + xcos(x) and g(x) = Sin(x) than find value of limx → 0⁡ ^f(x)⁄g(x)

(a) 2

(b) 1

(c) 3

(d) 4

I got this question at a job interview.

The question is from Indeterminate Forms in division Differential Calculus of Engineering Mathematics

Answer 1

Correct option is (a) 2

The best I can explain: \(\lim_{x\rightarrow 0}\frac{f(x)}{g(x)}=\lim_{x\rightarrow 0}\frac{e^x+xcos(x)}{sin(x)}=\lim_{x\rightarrow 0}\frac{e^x}{sin(x)}+\lim_{x\rightarrow 0}\frac{xcos(x)}{sin(x)}\)=1/0+0/0, i.e both are indeterminate

i.e., applying L’Hospital Rule

\(\lim_{x\rightarrow 0}\frac{e^x}{sin(x)}=\lim_{x\rightarrow 0}\frac{e^x}{cos(x)}+\lim_{x\rightarrow 0}\frac{cos(x)-xsin(x)}{cos(x)}\)=1+1=2