First slide

Introduction to limits

Question

The integer n for which $\underset{x \to 0}{L t} \frac{(\cos x - 1) (\cos x - e^{x})}{x^{n}}$ is a finite nonzero number is

Moderate

A

1
B

2
C

3
D

4

Solution

$(\cos x - 1) (\cos x - e^{x}) = ((1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdot \cdot \cdot) - 1) [(1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdot \cdot \cdot) - (1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} \cdot \cdot \cdot)]$
$= x^{2} (- \frac{1}{2} + \frac{x^{2}}{4!} \cdot \cdot \cdot) (- x - x^{2} - \frac{x^{3}}{3!} \cdot \cdot \cdot) = x^{3} (- \frac{1}{2} + \frac{x^{2}}{4!} \cdot \cdot \cdot) (- 1 - x - \frac{x^{2}}{3!} \cdot \cdot \cdot)$
$\frac{(\cos x - 1) (\cos x - e^{x})}{x^{3}} = (\frac{1}{2} + terms containing x and higher powers of x)$
$\underset{x \to 0}{L t} \frac{(\cos x - 1) (\cos x - e^{x})}{x^{3}} = \frac{1}{2} \Rightarrow n = 3.$

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