Question

Evaluate \(\lim_{x\rightarrow\infty}(sin(\frac{1}{x})+cos(\frac{1}{x}))^x\)

(a) 1

(b) e

(c) 0

(d) e^2

The question was asked during a job interview.

The origin of the question is Limits and Derivatives of Several Variables topic in chapter Partial Differentiation of Engineering Mathematics

Answer 1

Right choice is (b) e

For explanation I would say: \(\lim_{x\rightarrow\infty}(sin(\frac{1}{x})+cos(\frac{1}{x}))^x\)

Putting x=1/y,

\(\Rightarrow \lim_{y\rightarrow 0}(sin(y)+cos(y))^{\frac{1}{y}}\)

\(\Rightarrow \lim_{y\rightarrow 0}((y-\frac{y^3}{3!}+\frac{y^5}{5!}-…)+(1-\frac{y^2}{2!}+\frac{y^4}{4!}-….))^{\frac{1}{y}}\)

Neglecting higher powers of y,(as y is limits to 0 which is very small hence higher power terms can be neglected)

\(\Rightarrow\lim_{y\rightarrow 0}(1+y)^{\frac{1}{y}}\)

=>e