Question

Find the value of \(\lim_{x\rightarrow \infty}(1+\frac{1}{x})^x\)

(a) e-1

(b) e

(c) e + 1

(d) 1

This question was addressed to me at a job interview.

I want to ask this question from Indeterminate Forms in division Differential Calculus of Engineering Mathematics

Answer 1

Correct option is (b) e

The best I can explain: \(\lim_{x\rightarrow \infty}(1+\frac{1}{x})^x\)=1^∞ (Indeterminate form)

Hence

Let t=1/x

->y=\(\lim_{x\rightarrow\infty}(1+\frac{1}{x})^x=\lim_{t\rightarrow 0}(1+t)^{1/t}\)

Taking log on both side,

ln(y) = \(\lim_{t\rightarrow 0} \frac{ln(1+t)}{t}\) = 0/0 (Indeterminate form)

Applying L’Hospital Rule again

ln(y) = \(\lim_{t\rightarrow 0} \frac{1}{1+t}\) = 1

=> y = \(\lim_{x\rightarrow\infty}(1+\frac{1}{x})^x\) = e