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 nonzero number is

Moderate

A

1
B

2
C

3
D

4

Solution

Given that, $\lim_{x \to 0} \frac{(\cos x - 1) (\cos x - e^{x})}{x^{n}} =$ finite non zero number = $\lim_{x \to 0} \frac{(\cos x - 1) (1 + \cos x) (e^{x} - \cos x)}{x^{n} (1 + \cos x)}$ = $\lim_{x \to 0} (\frac{\sin^{2} x}{x^{2}}) . (\frac{e^{x} - \cos x}{x^{n - 2}}) . (\frac{1}{1 + \cos x})$ $= 1^{2} . \frac{1}{2} \lim_{x \to 0} \frac{[1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + .... \infty] - [1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \frac{x^{6}}{6!} + .... \infty]}{x^{n - 2}}$ $= \frac{1}{2} \lim_{x \to 0} \frac{(1 + x + \frac{x^{2}}{3!} + \frac{2 x^{3}}{4!} + ..... \infty)}{x^{n - 3}}$ for this limit to be finite n-3 = 0 $\Rightarrow$ n = 3

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