Limit Continuity and Differentiability 1 Question 13

14. The integer $n$ for which $\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^{x}\right)}{x^{n}}$ is a finite non-zero number, is

(2002, 2M)

(a) 1

(b) 2

(c) 3

(d) 4

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Answer:

Correct Answer: 14. (c)

Solution:

  1. $\lim _{x \rightarrow 0} \frac{1+5 x^{2}}{1+3 x^{2}}{ }^{1 / x^{2}}=\frac{\lim _{x \rightarrow 0}\left[\left(1+5 x^{2}\right)^{1 / 5 x^{2}}\right]^{5}}{\lim _{x \rightarrow 0}\left[\left(1+3 x^{2}\right)^{1 / 3 x^{2}}\right]^{3}}=\frac{e^{5}}{e^{3}}=e^{2}$

$$ =e^{\lim _{x \rightarrow \infty} \frac{5(x+4)}{x+1}}=e^{5} $$



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