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Limits And Derivatives

Multiple Choice Questions

31.

What is the value of $\lim_{x \to 0} \frac{\sin (x °)}{\tan (3 x °)}$

$\frac{1}{4}$
$\frac{1}{3}$
$\frac{1}{2}$
1

32.

What is $\lim_{x \to 0} \frac{3^{x} + 3^{- x} - 2}{x} = ?$

0
- 1
1
limit does not exist

33.

What is the maximum value of $\sin (x) \cos (x)$ ?

2
1
$\frac{1}{2}$
$2 \sqrt{2}$

34.

The value of ordinate of the graph of y = 2 + cos(x) lies in the interval

[0, 1]
[0, 3]
[ - 1, 1]
[1, 3]

35.

What is $\lim_{x \to 0} \frac{\sin (x) \log (1 - x)}{x^{2}} = ?$

- 1
Zero
- e
- $\frac{1}{e}$

36.
If $\lim_{x \to 1} \frac{x^{4} - 1}{x - 1} = \lim_{x \to k} \frac{x^{3} - k^{3}}{x^{2} - k^{2}} where k \neq 0, then what is the value of k ?$

$\frac{2}{3}$

$\frac{4}{3}$

$\frac{8}{3}$

$4$

$\frac{8}{3}$

$\lim_{x \to 1} \frac{x^{4} - 1}{x - 1} = \lim_{x \to k} \frac{x^{3} - k^{3}}{x^{2} - k^{2}} \lim_{x \to 1} \frac{x^{4} - 1}{x - 1} = \lim_{x \to 1} \frac{(x - 1) (x + 1) (x^{2} + 1)}{x - 1} = 4 And \lim_{x \to k} \frac{x^{3} - k^{3}}{x^{2} - k^{2}} = \lim_{x \to k} \frac{(x - k) (x^{2} + k^{2} + xk)}{(x - k)} = \frac{3}{2} k But it is given that - \lim_{x \to 1} \frac{x^{4} - 1}{x - 1} = \lim_{x \to k} \frac{x^{3} - k^{3}}{x^{2} - k^{2}} ∴ \frac{3}{2} k = 4 k = \frac{8}{3}$