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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

34.

The value of $\lim_{x \to 0} {(\frac{1 + 5 x^{2}}{1 + 3 x^{2}})}^{\frac{1}{x^{2}}}$

e²
e
$\frac{1}{e}$
$\frac{1}{e^{2}}$

35.

If f(5)=7 and f'(5)=7 then $\lim_{x \to 5} \frac{xf (5) - 5 f (x)}{x - 5}$ is given by

35
- 35
28
- 28

36.
The value of f(0) so that the function $\frac{1 - \cos (1 - \cos (x))}{x^{4}}$ is continuous everywhere is

$\frac{1}{2}$

$\frac{1}{4}$

$\frac{1}{6}$

$\frac{1}{8}$

$\frac{1}{8}$

$\lim_{x \to 0} \frac{1 - \cos (1 - \cos (x))}{x^{4}} = \lim_{x \to 0} \frac{2 (\frac{\sin^{2} (\frac{x}{2})}{2})}{x^{4}} = 2 \lim_{x \to 0} \frac{\sin^{4} (\frac{x}{2})}{{(\frac{x}{2})}^{4} 2^{4}} = \frac{1}{2^{3}} = \frac{1}{8}$

37.

$\lim_{x \to 0} \frac{\sin |x|}{x}$ is equal to

1
0
positive infinity
does not exist

38.

If y = $\tan^{- 1} \sqrt{\frac{1 - \sin (x)}{1 + \sin (x)}}$ , then the value of $\frac{dy}{dx}$ at x = $\frac{π}{6}$ is

- $\frac{1}{2}$
$\frac{1}{2}$
1
- 1

39.

The value of the limit $\lim_{x \to 1} \frac{\sin (e^{x - 1} - 1)}{\log (x)}$

0
e
$\frac{1}{e}$
1

40.

Let f(x) = $\frac{\sqrt{x + 3}}{x + 1}$ , then the value of $\lim_{x \to - 3 - 0} f (x)$ is

0
does not exist
$\frac{1}{2}$
$- \frac{1}{2}$