Continuity of Specific Functions
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Limits, Continuity and Differentiability

80110 If \(f(x)=\frac{|x-2|}{x-2}, \quad\) for \(\quad x \neq 2\)
\(=1, \quad \text { for } \quad x=2,\)
then which of the following statements is true?

1 \(\lim _{x \rightarrow 2^{+}} f(x)=f(2)\)
2 \(f(x)\) is discontinuous at \(x=2\)
3 \(\lim _{x \rightarrow 2^{-}} f(x)=f(2)\)
4 \(f(x)\) is continuous at \(x=2\)
MHT CET-2020
Limits, Continuity and Differentiability

80111 The function \(f(x)=\sqrt{x-2}\) is continuous in

1 \([2, \infty)\)
2 \((-\infty, 2]\)
3 \((-2, \infty)\)
4 \((-\infty, 2)\)
MHT CET-2019
Limits, Continuity and Differentiability

80112 If function
\(\begin{aligned} \mathbf{f}(\mathbf{x}) & =\mathbf{x}-\frac{|\mathbf{x}|}{\mathbf{x}}, & & \mathbf{x}\lt \mathbf{0} \\ & =\mathbf{x}+\frac{|\mathbf{x}|}{\mathbf{x}}, & & \mathbf{x}>\mathbf{0} \\ & =1, & & \mathbf{x}=\mathbf{0}\end{aligned}\)
then,

1 \(\lim _{x \rightarrow 0^{-}} f(x)\) does not exist
2 \(\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x)\)
3 \(\lim _{x \rightarrow 0^{+}}\)does not exist
4 \(f(x)\) is continuous at \(x=0\)
MHT CET-2019
Limits, Continuity and Differentiability

80113 If \(f(x)=\frac{a x+b}{x+1}\) and \(f(0)=4, f(1)=3\), then \(\mathbf{f}(\mathbf{2})=\)

1 1
2 0
3 \(\frac{8}{3}\)
4 \(\frac{7}{3}\)
MHT CET-2009
Limits, Continuity and Differentiability

80114 If
\(f(x)=\left\{\begin{array}{cl}1+(|\sin x|)^{\frac{a}{|\sin x|}}, & -\frac{\pi}{6}\lt x\lt 0 \\ b, & x=0 \\ e^{\frac{\tan 2 x}{\tan 3 x}}, & 0\lt x\lt \frac{\pi}{6}\end{array}\right.\) is continuous at \(x=0\), then the values of \(a\) and \(b\) are respectively.

1 \(\frac{3}{2}, \mathrm{e}^{3 / 2}\)
2 \(-\frac{2}{3}, \mathrm{e}^{-3 / 2}\)
3 \(\frac{2}{3}, \mathrm{e}^{2 / 3}\)
4 None of these
MHT CET-2011
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Limits, Continuity and Differentiability

80110 If \(f(x)=\frac{|x-2|}{x-2}, \quad\) for \(\quad x \neq 2\)
\(=1, \quad \text { for } \quad x=2,\)
then which of the following statements is true?

1 \(\lim _{x \rightarrow 2^{+}} f(x)=f(2)\)
2 \(f(x)\) is discontinuous at \(x=2\)
3 \(\lim _{x \rightarrow 2^{-}} f(x)=f(2)\)
4 \(f(x)\) is continuous at \(x=2\)
MHT CET-2020
Limits, Continuity and Differentiability

80111 The function \(f(x)=\sqrt{x-2}\) is continuous in

1 \([2, \infty)\)
2 \((-\infty, 2]\)
3 \((-2, \infty)\)
4 \((-\infty, 2)\)
MHT CET-2019
Limits, Continuity and Differentiability

80112 If function
\(\begin{aligned} \mathbf{f}(\mathbf{x}) & =\mathbf{x}-\frac{|\mathbf{x}|}{\mathbf{x}}, & & \mathbf{x}\lt \mathbf{0} \\ & =\mathbf{x}+\frac{|\mathbf{x}|}{\mathbf{x}}, & & \mathbf{x}>\mathbf{0} \\ & =1, & & \mathbf{x}=\mathbf{0}\end{aligned}\)
then,

1 \(\lim _{x \rightarrow 0^{-}} f(x)\) does not exist
2 \(\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x)\)
3 \(\lim _{x \rightarrow 0^{+}}\)does not exist
4 \(f(x)\) is continuous at \(x=0\)
MHT CET-2019
Limits, Continuity and Differentiability

80113 If \(f(x)=\frac{a x+b}{x+1}\) and \(f(0)=4, f(1)=3\), then \(\mathbf{f}(\mathbf{2})=\)

1 1
2 0
3 \(\frac{8}{3}\)
4 \(\frac{7}{3}\)
MHT CET-2009
Limits, Continuity and Differentiability

80114 If
\(f(x)=\left\{\begin{array}{cl}1+(|\sin x|)^{\frac{a}{|\sin x|}}, & -\frac{\pi}{6}\lt x\lt 0 \\ b, & x=0 \\ e^{\frac{\tan 2 x}{\tan 3 x}}, & 0\lt x\lt \frac{\pi}{6}\end{array}\right.\) is continuous at \(x=0\), then the values of \(a\) and \(b\) are respectively.

1 \(\frac{3}{2}, \mathrm{e}^{3 / 2}\)
2 \(-\frac{2}{3}, \mathrm{e}^{-3 / 2}\)
3 \(\frac{2}{3}, \mathrm{e}^{2 / 3}\)
4 None of these
MHT CET-2011
For More Updates join with Us
Limits, Continuity and Differentiability

80110 If \(f(x)=\frac{|x-2|}{x-2}, \quad\) for \(\quad x \neq 2\)
\(=1, \quad \text { for } \quad x=2,\)
then which of the following statements is true?

1 \(\lim _{x \rightarrow 2^{+}} f(x)=f(2)\)
2 \(f(x)\) is discontinuous at \(x=2\)
3 \(\lim _{x \rightarrow 2^{-}} f(x)=f(2)\)
4 \(f(x)\) is continuous at \(x=2\)
MHT CET-2020
Limits, Continuity and Differentiability

80111 The function \(f(x)=\sqrt{x-2}\) is continuous in

1 \([2, \infty)\)
2 \((-\infty, 2]\)
3 \((-2, \infty)\)
4 \((-\infty, 2)\)
MHT CET-2019
Limits, Continuity and Differentiability

80112 If function
\(\begin{aligned} \mathbf{f}(\mathbf{x}) & =\mathbf{x}-\frac{|\mathbf{x}|}{\mathbf{x}}, & & \mathbf{x}\lt \mathbf{0} \\ & =\mathbf{x}+\frac{|\mathbf{x}|}{\mathbf{x}}, & & \mathbf{x}>\mathbf{0} \\ & =1, & & \mathbf{x}=\mathbf{0}\end{aligned}\)
then,

1 \(\lim _{x \rightarrow 0^{-}} f(x)\) does not exist
2 \(\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x)\)
3 \(\lim _{x \rightarrow 0^{+}}\)does not exist
4 \(f(x)\) is continuous at \(x=0\)
MHT CET-2019
Limits, Continuity and Differentiability

80113 If \(f(x)=\frac{a x+b}{x+1}\) and \(f(0)=4, f(1)=3\), then \(\mathbf{f}(\mathbf{2})=\)

1 1
2 0
3 \(\frac{8}{3}\)
4 \(\frac{7}{3}\)
MHT CET-2009
Limits, Continuity and Differentiability

80114 If
\(f(x)=\left\{\begin{array}{cl}1+(|\sin x|)^{\frac{a}{|\sin x|}}, & -\frac{\pi}{6}\lt x\lt 0 \\ b, & x=0 \\ e^{\frac{\tan 2 x}{\tan 3 x}}, & 0\lt x\lt \frac{\pi}{6}\end{array}\right.\) is continuous at \(x=0\), then the values of \(a\) and \(b\) are respectively.

1 \(\frac{3}{2}, \mathrm{e}^{3 / 2}\)
2 \(-\frac{2}{3}, \mathrm{e}^{-3 / 2}\)
3 \(\frac{2}{3}, \mathrm{e}^{2 / 3}\)
4 None of these
MHT CET-2011
NEET DIGITAL LIBRARY
Limits, Continuity and Differentiability

80110 If \(f(x)=\frac{|x-2|}{x-2}, \quad\) for \(\quad x \neq 2\)
\(=1, \quad \text { for } \quad x=2,\)
then which of the following statements is true?

1 \(\lim _{x \rightarrow 2^{+}} f(x)=f(2)\)
2 \(f(x)\) is discontinuous at \(x=2\)
3 \(\lim _{x \rightarrow 2^{-}} f(x)=f(2)\)
4 \(f(x)\) is continuous at \(x=2\)
MHT CET-2020
Limits, Continuity and Differentiability

80111 The function \(f(x)=\sqrt{x-2}\) is continuous in

1 \([2, \infty)\)
2 \((-\infty, 2]\)
3 \((-2, \infty)\)
4 \((-\infty, 2)\)
MHT CET-2019
Limits, Continuity and Differentiability

80112 If function
\(\begin{aligned} \mathbf{f}(\mathbf{x}) & =\mathbf{x}-\frac{|\mathbf{x}|}{\mathbf{x}}, & & \mathbf{x}\lt \mathbf{0} \\ & =\mathbf{x}+\frac{|\mathbf{x}|}{\mathbf{x}}, & & \mathbf{x}>\mathbf{0} \\ & =1, & & \mathbf{x}=\mathbf{0}\end{aligned}\)
then,

1 \(\lim _{x \rightarrow 0^{-}} f(x)\) does not exist
2 \(\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x)\)
3 \(\lim _{x \rightarrow 0^{+}}\)does not exist
4 \(f(x)\) is continuous at \(x=0\)
MHT CET-2019
Limits, Continuity and Differentiability

80113 If \(f(x)=\frac{a x+b}{x+1}\) and \(f(0)=4, f(1)=3\), then \(\mathbf{f}(\mathbf{2})=\)

1 1
2 0
3 \(\frac{8}{3}\)
4 \(\frac{7}{3}\)
MHT CET-2009
Limits, Continuity and Differentiability

80114 If
\(f(x)=\left\{\begin{array}{cl}1+(|\sin x|)^{\frac{a}{|\sin x|}}, & -\frac{\pi}{6}\lt x\lt 0 \\ b, & x=0 \\ e^{\frac{\tan 2 x}{\tan 3 x}}, & 0\lt x\lt \frac{\pi}{6}\end{array}\right.\) is continuous at \(x=0\), then the values of \(a\) and \(b\) are respectively.

1 \(\frac{3}{2}, \mathrm{e}^{3 / 2}\)
2 \(-\frac{2}{3}, \mathrm{e}^{-3 / 2}\)
3 \(\frac{2}{3}, \mathrm{e}^{2 / 3}\)
4 None of these
MHT CET-2011
Join With Us for More Updates
Limits, Continuity and Differentiability

80110 If \(f(x)=\frac{|x-2|}{x-2}, \quad\) for \(\quad x \neq 2\)
\(=1, \quad \text { for } \quad x=2,\)
then which of the following statements is true?

1 \(\lim _{x \rightarrow 2^{+}} f(x)=f(2)\)
2 \(f(x)\) is discontinuous at \(x=2\)
3 \(\lim _{x \rightarrow 2^{-}} f(x)=f(2)\)
4 \(f(x)\) is continuous at \(x=2\)
MHT CET-2020
Limits, Continuity and Differentiability

80111 The function \(f(x)=\sqrt{x-2}\) is continuous in

1 \([2, \infty)\)
2 \((-\infty, 2]\)
3 \((-2, \infty)\)
4 \((-\infty, 2)\)
MHT CET-2019
Limits, Continuity and Differentiability

80112 If function
\(\begin{aligned} \mathbf{f}(\mathbf{x}) & =\mathbf{x}-\frac{|\mathbf{x}|}{\mathbf{x}}, & & \mathbf{x}\lt \mathbf{0} \\ & =\mathbf{x}+\frac{|\mathbf{x}|}{\mathbf{x}}, & & \mathbf{x}>\mathbf{0} \\ & =1, & & \mathbf{x}=\mathbf{0}\end{aligned}\)
then,

1 \(\lim _{x \rightarrow 0^{-}} f(x)\) does not exist
2 \(\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x)\)
3 \(\lim _{x \rightarrow 0^{+}}\)does not exist
4 \(f(x)\) is continuous at \(x=0\)
MHT CET-2019
Limits, Continuity and Differentiability

80113 If \(f(x)=\frac{a x+b}{x+1}\) and \(f(0)=4, f(1)=3\), then \(\mathbf{f}(\mathbf{2})=\)

1 1
2 0
3 \(\frac{8}{3}\)
4 \(\frac{7}{3}\)
MHT CET-2009
Limits, Continuity and Differentiability

80114 If
\(f(x)=\left\{\begin{array}{cl}1+(|\sin x|)^{\frac{a}{|\sin x|}}, & -\frac{\pi}{6}\lt x\lt 0 \\ b, & x=0 \\ e^{\frac{\tan 2 x}{\tan 3 x}}, & 0\lt x\lt \frac{\pi}{6}\end{array}\right.\) is continuous at \(x=0\), then the values of \(a\) and \(b\) are respectively.

1 \(\frac{3}{2}, \mathrm{e}^{3 / 2}\)
2 \(-\frac{2}{3}, \mathrm{e}^{-3 / 2}\)
3 \(\frac{2}{3}, \mathrm{e}^{2 / 3}\)
4 None of these
MHT CET-2011