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.

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.

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

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.

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

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

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

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

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.

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

4. none of these