Question

Find n for which \(\lim_{x\rightarrow 0}\frac{(cos(x)-1)(cos(x)-e^x)}{x^n}\), has non zero value.

(a) >=1

(b) >=2

(c) <=2

(d) ~2

The question was posed to me during an internship interview.

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

Answer 1

Correct choice is (b) >=2

For explanation: \(\lim_{x\rightarrow 0}\frac{(cos(x)-1)(cos(x)-e^x)}{x^n}=(0/0)\)

By L’Hospital Rule two times we get

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

Hence, limit have non zero limit, if n ≠ 0 and (n-1) ≠ 0 and (n-2) >= 0 means n >= 2.