Question

What is the value of \(\lim\limits_{x \rightarrow 0}\frac{e^x(sin^2⁡ x)}{x^3}\)?

(a) 2

(b) 3

(c) 1

(d) 0

I have been asked this question in unit test.

The origin of the question is Limits of Trigonometric Functions in portion Limits and Derivatives of Mathematics – Class 11

Answer 1

The correct choice is (c) 1

The explanation: \(\lim\limits_{x \rightarrow 0}\frac{sin^2⁡ x}{x^2}\) x \(\lim\limits_{x \rightarrow 0}\frac{e^x}{x}\)

We apply L’Hospital’s rule and differentiate numerator and denominator.

= 1 x \(\lim\limits_{x \rightarrow 0}\frac{e^x}{1}\)

= 1