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Limits and Derivatives

Short Answer Type

31.

If N = n!(n $(n \in N, n > 2)$ , then find

$\lim_{N \to \infty} [{(\log_{2} (N))}^{- 1} + (\log_{3} {(N)}^{- 1}) + . . . + \log_{n} {(N)}^{- 1}]$

32.

Use the formula $\lim_{x \to 0} \frac{a^{x} - 1}{x} = \log_{e} (a) to compute \lim_{x \to 0} \frac{2^{x} - 1}{\sqrt{1 + x} - 1}$

Multiple Choice Questions

33.
The value of $\lim_{x \to 0} \frac{\sin (x) + \cos (x) - 1}{x^{2}}$

1

$\frac{1}{2}$

$\frac{- 1}{2}$

0

$\frac{1}{2}$

$\lim_{x \to 0} \frac{\sin^{2} (x) + \cos (x) - 1}{x^{2}} = \lim_{x \to 0} \frac{\cos (x) - \cos^{2} (x)}{x^{2}} = \lim_{x \to 0} (\frac{1 - \cos (x)}{x^{2}}) = \lim_{x \to 0} \frac{2 \sin^{2} (\frac{x}{2})}{x^{2}} \cos (x) = \lim_{x \to 0} \frac{2 \sin^{2} (\frac{x}{2})}{{(\frac{x}{2})}^{2} . 4} . \lim_{x \to 0} \cos (x) = \frac{2}{4} \times 1 = \frac{1}{2}$