Single Point Continuity

Q.

A function f(x) is continuous at a point x = a, then which of the following is incorrect.

1. 

2. 

3. 

4. 

Q. The area of the triangle on the complex plane formed by the complex numbers Z, - iZ and Z+iZ is

1. 
2. 
3. 
4. 

Q.

$\underset{x\to 0}{lim}\frac{lo{g}_e(1+x)}{x}=$

1. 

2. e

3. 1

4. -1

Q. If a, b, y are the altitudes of a $△ABC$ and 2s denotes its perimeter,
then $a^{-1}+B^{-1}+y^{-1}$ is equal to

1. 

2. 

3. 

4. None of these

Q. Let $f\left(x\right)=[3+2\cos x],x\in (-\frac{\pi }{2},\frac{\pi }{2})$, where [.] denotes the greatest integer function. The number of points of discontinuity of $f\left(x\right)$ is

1. $3$
2. $2$
3. $5$
4. $Noneofthese$

Q. If the function
$f\left(x\right)=\{\begin{array}{cc}(1+|sinx|{)}^{\frac{a}{|sinx|,}},& -\frac{\pi }{6}<x<0\\ b,& x=0\\ e^{\frac{tan2x}{tan3x}},& 0<x<\frac{\pi }{6}\end{array}$, is continuous at x = 0, then

1. $a=lo{g}_{e}b,b=2$
2. $a=lo{g}_{e}b,a=\frac{2}{3}$
3. none of these
4. $b=lo{g}_{e}a,a=\frac{2}{3}$

Q.

find the domain and range of the real function f defined by $f\left(x\right)=\sqrt{x-1}$

Or

Let f= {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from Z to Z defined by f(x) = ax+b from some integers a and b. Determine a and b.

Q.

If the function
$f\left(x\right)=\{\begin{array}{cc}(1+|sinx|{)}^{\frac{a}{|sinx|,}},& -\frac{\pi }{6}<x<0\\ b,& x=0\\ e^{\frac{tan2x}{tan3x}},& 0<x<\frac{\pi }{6}\end{array}$, is continuous at x = 0, then

1. none of these

2. $a=lo{g}_{e}b,b=2$

3. $a=lo{g}_{e}b,a=\frac{2}{3}$

4. $b=lo{g}_{e}a,a=\frac{2}{3}$

Q. Given $f\left(x\right)=|x-1|+|x+1|$. Then f(x) is

1. Discontinuous at x = 0

2. discontinuous at x = ± 1

3. Discontinuous at all integers

4. continuous everywhere

Q.

A function $f\left(x\right)$ is continuous at a point $x=a$, then which of the following is incorrect.

1. $f(a+0.0001)=f(a-0.0001)$

2. $f\left(a\right)=\underset{x\to a^{+}}{lim}f\left(x\right)$

3. $\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)$

4. $\underset{h\to 0}{lim}f(a-h)=f\left(a\right)$

Q.

A function $f\left(x\right)$ is continuous at a point $x=a$, then which of the following is incorrect.

1. $f(a+0.0001)=f(a-0.0001)$

2. $f\left(a\right)=\underset{x\to a^{+}}{lim}f\left(x\right)$

3. $\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)$

4. $\underset{h\to 0}{lim}f(a-h)=f\left(a\right)$

Q. $\underset{x\to a-}{lim}f\left(x\right)=11$ and f(x)=2x+3. If f(x) is a continuous function, the value of a is .

1. 4
2. 7
3. 8
4. 11

Q. A continuous function can have some points where limit does not exist.

1. True
2. False

Q. If the function $f\left(x\right)=2p[2x+5]+q[3x-7]$ is continuous at $x=1$, then
$($where [.] denotes greatest integer function and $p,qϵR)$

1. $2p+q=0$
2. $p+2q=0$
3. $p+q=0$
4. $10p-7q=0$

Q.

Find the $li{m}_{x\to 1}[\frac{x-2}{x^2-x}-\frac{1}{x^3-3x^2+2x}]$

Q.

A function $f\left(x\right)$ is continuous at a point $x=a$, then which of the following is incorrect.

1. $f(a+0.0001)=f(a-0.0001)$

2. $f\left(a\right)=\underset{x\to a^{+}}{lim}f\left(x\right)$

3. $\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)$

4. $\underset{h\to 0}{lim}f(a-h)=f\left(a\right)$

Q. Find the derivative of the following function from first principle:
$-x$

Q. Let $f\left(x\right)=\{\begin{array}{cc}\frac{\tan kx}{x},& x<0\\ 3x+2k^2,& x\ge 0\end{array}$. If $f\left(x\right)$ is continuous at $x=0$, then number of values of $k$ is

1. 1
2. 2
3. more than 2
4. none