First slide

Introduction to limits

Question

The integral value of n for which $\lim_{x \to 0} \frac{\cos^{2} x - \cos x - e^{x} \cos x + e^{x} - (\frac{x^{3}}{2})}{x^{n}}$ is finite and non zero is

Moderate

A

2
B

4
C

5
D

6

Solution

Given $\lim_{x \to 0} \frac{\cos^{2} x - \cos x - e^{x} \cos x + e^{x} - \frac{x^{3}}{2}}{x^{n}}$ $= \lim_{x \to 0} \frac{(\cos x - 1) (\cos x - e^{x}) - \frac{x^{3}}{2}}{x^{n}}$
$= \lim_{x \to 0} \frac{(1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + .... - 1)}{x^{n}}$ $\times \frac{[(1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - .. .) - (1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} - ...)] - \frac{x^{3}}{2}}{x^{n}}$ $\lim_{x \to 0} \frac{(- \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} ....) [(- x - x^{2} - \frac{x^{3}}{3!} - \frac{2 x^{5}}{5!} - ....)] - \frac{x^{3}}{2}}{x^{n}}$
$\lim_{x \to 0} \frac{(\frac{x^{3}}{2} + \frac{x^{4}}{2} + \frac{x^{5}}{12} - \frac{x^{5}}{24} + .0. .) - \frac{x^{3}}{2}}{x^{n}}$
= non zero if n=4

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