Single Point Continuity

Q.

The derivative of $\log \left|x\right|$ is

1. $\frac{1}{x},x>0$

2. $\frac{1}{\left|x\right|},x\ne 0$

3. $\frac{1}{x},x\ne 0$

4. None of these

Q.

The function $\mathrm{f}\left(\mathrm{x}\right)=\log \left(1+\mathrm{x}\right)-\left(\frac{2\mathrm{x}}{\left(2+\mathrm{x}\right)}\right)$ is increasing for all values of $\mathrm{x}$, then

1. $\left(0,\infty \right)$

2. $\left(-\infty ,0\right)$

3. $\left(-\infty ,\infty \right)$

4. None of these

Q. $\underset{x\to 0}{lim}\frac{{\sin }^2\left(\pi {\cos }^{4}x\right)}{x^4}$ is equal to:

1. $4\pi$
2. ${\pi }^2$
3. $4{\pi }^2$
4. $2{\pi }^2$

Q. The value of $\underset{x\to 0}{lim}\left[\frac{\sin [x-3]}{[x-3]}\right]$ is
(where $[.]$ represents the greatest integer function)

1. $0$
2. $1$
3. $\sin 1$
4. does not exist

Q. If $f\left(x\right)=\left[x\right]-\left[\frac{x}{4}\right],x\in R$, where $\left[x\right]$ denotes the greatest integer function, then :

1. $\underset{x\to 4^{+}}{lim}f\left(x\right)$ exists but $\underset{x\to 4^{-}}{lim}f\left(x\right)$ does not exist.
2. $\underset{x\to 4^{-}}{lim}f\left(x\right)$ exists but $\underset{x\to 4^{+}}{lim}f\left(x\right)$ does not exist.
3. Both $\underset{x\to 4^{-}}{lim}f\left(x\right)$ and $\underset{x\to 4^{+}}{lim}f\left(x\right)$ exist but are not equal.
4. $f$ is continuous at $x=4$.

Q. Evaluate the given limit :
$\underset{x\to 0}{lim}\frac{ax+x\cos x}{b\sin x}$

Q.

The function $\mathrm{f}\left(\mathrm{x}\right)=\frac{\ln \left(1+\mathrm{ax}\right)-\ln (1-\mathrm{bx})}{\mathrm{x}}$ is not defined at $\mathrm{x}=0$. The value which should be assigned to $\mathrm{f}$ at $\mathrm{x}=0$, so that it is continuous at $\mathrm{x}=0$, is

1. $\mathrm{a}-\mathrm{b}$

2. $\mathrm{a}+\mathrm{b}$

3. $\ln \mathrm{a}-\ln \mathrm{b}$

4. none of these

Q. Find the derivative of $f\left(x\right)=10x$.

Q. If the function $f\left(x\right)=\{\begin{array}{cc}a|\pi -x|+1,& x\le 5\\ & \\ b|x-\pi |+3,& x>5\end{array}$

is continuous at $x=5$, then the value of $a-b$ is :

1. $\frac{2}{\pi -5}$
2. $\frac{2}{\pi +5}$
3. $\frac{2}{5-\pi }$
4. $\frac{-2}{\pi +5}$

Q. For what value of k, is the function $f\left(x\right)=\{\begin{array}{c}\frac{k\cos x}{\pi -2x},\text{if}x\ne \frac{\pi }{2}\\ 3\text{if}x=\frac{\pi }{2}\end{array}$ continuous for all real $x?$

1. $0$
2. $\pi$
3. $3$
4. $6$

Q.

How do you know if a function is continuous and differentiable?

Q.

The function $\mathrm{f}\left(\mathrm{x}\right)=\frac{\ln \left(\mathrm{\pi }+\mathrm{x}\right)}{\ln \left(\mathrm{e}+\mathrm{x}\right)}$ is

1. Increasing on $[0,\infty ]$

2. Decreasing on $(0,\infty )$

3. Decreasing on$[0,\frac{\mathrm{\pi }}{\mathrm{e}})$ and increasing on $[\frac{\mathrm{\pi }}{\mathrm{e}},\infty )$

4. Increasing on$[0,\frac{\mathrm{\pi }}{\mathrm{e}})$ and decreasing on $[\frac{\mathrm{\pi }}{\mathrm{e}},\infty )$

Q. Let $[.]$ and $\{.\}$ be the greatest integer function and fractional part function respectively. Then the number of points of discontinuity of the function $f\left(x\right)=\sin \left(\{2^x+\left[2^x\right]+\left[3^{–x}\right]\}\right)$ for $x\in [0,4]$ is

Q. Let $M$ and $m$ be respectively the absolute maximum and the absolute minimum value of the function, $f\left(x\right)=2x^3-9x^2+12x+5$ in the interval $[0,3]$. Then $M-m$ is equal to :

1. $5$
2. $1$
3. $4$
4. $9$

Q. If $\underset{x\to 0}{lim}(x^{-3}\sin 3x+a{x}^{-2}+b)=0$ , then $a+2b$ is equal to​​​​​​​

Q. If $f\left(x\right)$ is continuous and $f\left(\frac{9}{2}\right)=\frac{2}{9}$, then $\underset{x\to 0}{lim}f\left(\frac{1-\cos 3x}{x^2}\right)$ is equal to

1. $\frac{9}{2}$
2. $\frac{2}{9}$
3. $0$
4. $\frac{8}{9}$

Q. Find the relationship between a and b so that the function f defined by is continuous at x = 3.

Q. Find the derivative of $f\left(x\right)=\frac{1}{x}$.

Q. Let $\left[x\right]$ be the greatest integer less than or equal to $x$. Then, at which of the following point(s) the function $f\left(x\right)=x\cos \left(\pi \right(x+\left[x\right]\left)\right)$ is discontinuous?

1. $x=-1$
2. $x=0$
3. $x=1$
4. $x=2$

Q. $\underset{x\to \infty }{lim}x\cos \left(\frac{\pi }{8x}\right)\sin \left(\frac{\pi }{8x}\right)=$

1. $\pi$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{8}$
4. $\frac{\pi }{4}$

Q.

1. $-24$
2. $37$
3. $42$
4. $-62$

Q. "The product of three consecutive positive integers is divisible by 6". Is.this statement true or false"? Justify your answer.

Q. Arrange the following limits in the ascending order of their values
$\left(1\right)\underset{x\to \infty }{lim}{\left(\frac{1+x}{2+x}\right)}^{x+2}$
$\left(2\right)\underset{x\to 0}{lim}{(1+2x)}^{3/x}$
$\left(3\right)\underset{\theta \to 0}{lim}\left(\frac{\sin \theta }{2\theta }\right)$
$\left(4\right)\underset{x\to 0}{lim}\frac{\ln (1+x)}{x}$

1. $1,2,3,4$
2. $1,3,4,2$
3. $1,4,3,2$
4. $3,4,1,2$

Q. Evaluate the limit:
$\underset{x\to 0}{lim}\left(\frac{a^x-a^{-x}}{x}\right)$

Q. $\underset{n\to \infty }{lim}{\left(\frac{n^2-n+1}{n^2-n-1}\right)}^{n(n-1)}$ is equal to

1. $e$
2. $e^2$
3. $e^{-1}$
4. $1$

Q. Is the function defined by continuous at x = π ?

Q. If the function $f\left(x\right)$ satisfies $\underset{x\to 1}{lim}\frac{f\left(x\right)-2}{x^2-1}=\pi$, evaluate $\underset{x\to 1}{lim}f\left(x\right)$

Q. If $f\left(x\right)=x\sin \left(\frac{1}{x}\right),x\ne 0$ is continuous at $x=0,$ then the value of $f\left(0\right)$ is

1. $1$
2. $0$
3. $2$
4. $-1$

Q.

What is $C$ in an integral?

Q. The value of $\underset{n\to \infty }{lim}(\sqrt{n^2+n+1}-\left[\sqrt{n^2+n+1}\right]);n\in Z,$ where $[.]$ denotes the greatest integer function is

1. $0$
2. $\frac{1}{2}$
3. $\frac{2}{3}$
4. $\frac{1}{4}$