First slide

Introduction to limits

Question

The integer n for which $\lim_{x \to 0} \frac{(\cos x - 1) (\cos x - e^{x})}{x^{n}}$ is a finite non zero number is

Moderate

A

1
B

2
C

3
D

4

Solution

$\lim_{x \to 0} \frac{\cos x - 1}{x^{2}} . \frac{\cos x - e^{x}}{x^{n - 2}}$ use 1. Hospital rule = $- \frac{1}{2} \lim_{x \to 0} \frac{\cos x - e^{x}}{x^{n - 2}} = - 1 / 2 \lim_{x \to 0} \frac{- \sin x - e^{x}}{(n - 2) x^{n - 3}} i s f i n i t e \Rightarrow n - 3 = 0 \Rightarrow n = 3$

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