1. Let f be a continuous function on satisfying
f (x + y) = f(x) + f(y) for all x, \[y \epsilon R\] and
f(1) = 5 then \[\lim_{x \rightarrow 4}f\left(x\right)\] is equal to

a) 4

b) 80

c) 0

d) None of these

Explanation:

2. Let \[f\left(x\right)=e^{x} sgn\left(x+\left[x\right]\right)\] , where sgn is the signum
function and [x] is the greatest integer function.
Then

a) \[\lim_{x \rightarrow 0+}f\left(x\right)=0\]

b) \[\lim_{x \rightarrow 0+}f\left(x\right)=-1\]

c) \[\lim_{x \rightarrow 0+}f\left(x\right)=1\]

d) \[\lim_{x \rightarrow 0+}f\left(x\right)=2\]

Explanation:

3. Let \[f\left(x\right)=\begin{cases}\frac{x^{2}+2\cos x-2}{x^{4}}, & x < 0\\\frac{\sin x-\log\left(e^{x}\cos x\right)}{6x^{2}}, & x > 0\end{cases}\]

The value of f(0), so that f is continuous is

a) \[\frac{1}{5}\]

b) \[\frac{1}{3}\]

c) \[\frac{1}{12}\]

d) \[\frac{1}{6}\]

Explanation:

4. If \[f\left(x\right)=x\left(e^{1/x}-1\right)\] , then \[\lim_{x \rightarrow \infty}f\left(x\right)\] is

a) -1

b) 1

c) 0

d) none of these

Explanation:

5. The value of f(0) so that the function \[f\left(x\right)=\frac{1-\cos\left(1-\cos x\right)}{x^{4}}\]

is continuous everywhere is

a) 1/8

b) 1/2

c) 1/4

d) none of these

Explanation:

6. The function \[f\left(x\right)=\frac{\log\left(1+ax\right)-\log\left(1-bx\right)}{x}\]

is not defined at x = 0. The value which should be
assigned to f at x = 0 so that it is continuous there, is

a) a- b

b) a +b

c) log a + log b

d) none of these

Explanation:

7. The function \[f\left(x\right)=\frac{1+\sin x-\cos x}{1-\sin x-\cos x }\]
is not defined at x = 0. The value of f (0) so that f (x)
is continuous at x = 0, is

a) 1

b) -1

c) 0

d) none of these

Explanation:

8. The function \[f\left(x\right)=\frac{\cos x-\sin x}{\cos 2x }\]
is not defined at \[ x=\pi/4\] . The value of \[ f\left(\pi/4\right)\] so
that f(x) is continuous everywhere, is

a) 1

b) -1

c) \[\sqrt{2}\]

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

Explanation:

9. The value of f(0), for \[f\left(x\right)=\left(1+\tan^{2}\sqrt{x}\right)^{1/2x}\] , so
that f(x) is continuous everywhere, is

a) e

b) 1/2

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

d) 0

Explanation:

10. The value of \[\lim_{x \rightarrow 0}\frac{\left(1+x\right)^{1/4}-\left(1-x\right)^{1/4}}{x}\] is

a) 1/2

b) 0

c) -1

d) -1/2

Explanation: