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:
- $\lim _{x \rightarrow 0} \frac{1+5 x^{2}}{1+3 x^{2}}{ }^{1 / x^{2}}=\frac{\lim _{x \rightarrow 0}\left[\left(1+5 x^{2}\right)^{1 / 5 x^{2}}\right]^{5}}{\lim _{x \rightarrow 0}\left[\left(1+3 x^{2}\right)^{1 / 3 x^{2}}\right]^{3}}=\frac{e^{5}}{e^{3}}=e^{2}$
$$ =e^{\lim _{x \rightarrow \infty} \frac{5(x+4)}{x+1}}=e^{5} $$
