1. \[\lim_{x \rightarrow0}\frac{\sin^{-1}x-\tan^{-1}x}{x^{3}}\] is equal to

a) 2

b) 1

c) -1

d) 1/2

Explanation:

2. \[\lim_{x \rightarrow0}\frac{e^{x^{2}}-\cos x}{x^{2}}\] is equal to

a) 3/2

b) 1/2

c) 2/3

d) none of these

Explanation:

3. \[\lim_{x \rightarrow-1+}\frac{\sqrt{\pi}-\sqrt{\cos^{-1}x}}{\sqrt{x+1}}\] is equal to

a) \[1/\sqrt{2}\]

b) \[1/\sqrt{2\pi}\]

c) \[1/\sqrt{\pi}\]

d) none of these

Explanation:

4. The value of f(0) so that the function \[f\left(x\right)=\frac{\cos\left(\sin x\right)-\cos x}{x^{4}}\] is continuous at each point in its domains , is equal to

a) 2

b) 1/6

c) 2/3

d) -1/3

Explanation: For f to be continuous, we must have

5. Let f(x) =\[\begin{cases}-2\sin x & if & x\leq-\pi/2 \\A\sin x+B & if & -\pi/2 < x<\pi/2\\\cos x & if & x\geq\pi/2\end{cases}\]

The values of A and B so that f(x) is continuous every where are

a) A = 0 ,B = 1

b) A = 1 ,B = 1

c) A = -1 ,B = 1

d) A = -1 ,B = 0

Explanation: Since sin x and cos x are continuous functions,

6. Let \[f\left(x\right)=\frac{\tan\left[e^{2}\right]x^{3}-\tan\left[-e^{2}\right]x^{3}}{\sin^{3}x},x\neq0\]

The value of f(0) for which f(x) is continuous is

a) 15

b) 12

c) -12

d) 14

Explanation:

7. The value of f(0) so that the function \[f\left(x\right)=\frac{\sqrt{1+x}-\sqrt[3]{1+x}}{x}\]

becomes continuous, is equal to

a) 1/6

b) 1/4

c) 2

d) 1/3

Explanation: The function f(x) is continuous except possibly

8. Let \[f\left(x\right)=\left(2-\frac{x}{a}\right)^{\tan\left(\frac{\pi x}{2a}\right)},x\neq a\]

The value which should be assigned to f at x = a so that it is continuous everywhere is

a) \[2/\pi\]

b) \[e^{-2/\pi}\]

c) 2

d) \[e^{2/\pi}\]

Explanation: For f to be continuous, we must have

9. Let \[f\left(x\right)=\begin{cases}\left(\frac{e^{-x}+x^{2}-1}{-x}\right)^{\frac{-1}{x}} & -1\leq x < 0\\\frac{e^{1/x}+e^{2/x}+e^{3/\mid x \mid}}{ae^{2/x}+be^{3/\mid x \mid}} & 0< x< 1\end{cases}\]

The value of b so that \[\lim_{x\rightarrow 0}f\left(x\right)\] exist is

a) \[e^{1/2}\]

b) \[e^{3/2}\]

c) \[e^{-3/2}\]

d) \[e^{2}\]

Explanation:

10. If the function \[f\left(x\right)=\begin{cases}\frac{Ax^{3}+x^{2}-\left(A+2\right)x+A}{x-2} & x \neq 2\\2 & x=2 \end{cases}\]

is continuous at x=2 , then

a) A = 0

b) A = 1

c) A = -1

d) none of these

Explanation: Since f is continuous at x = 2, the only possible