Question

Find domain of n for which limx → 0e^nxCot(nx), has non zero value.

(a) n ∈ (0,∞) ∩ (1,5)

(b) n ∈ (-∞,∞) ∩ (1,5)

(c) n ∈ (-∞,∞)

(d) n ∈ (-∞,∞) ~ 5

This question was addressed to me in my homework.

I need to ask this question from Limits and Derivatives of Several Variables in section Partial Differentiation of Engineering Mathematics

Answer 1

Correct option is (c) n ∈ (-∞,∞)

The explanation is: \(\lim_{x\rightarrow 1}\frac{e^{nx}cos(nx)}{sin(nx)}=(1/0)\)

By L’hospital Rule we get

\(\Rightarrow \lim_{x\rightarrow 0}\frac{ne^{nx}(-sin(nx)+cos(nx))}{ncos(nx)}=n/n=1\)

Hence domain of n is n ∈ (-∞,∞).