## Limits, Continuity and Differentiability Questions and Answers Part-8

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

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