Download App

Continuity — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsContinuity1 Mark
MCQ
The function $\text{f(x)=}\begin{cases}\frac{\text{e}\frac{1}{\text{x}}-1}{\text{e}\frac{1}{\text{x}}+1},&\text{x}\neq0\\0,&\text{x}=0\end{cases}$
  • A
    is continuous at $x = 0$
  • ✓
    is not continuous at $x = 0$
  • C
    is not continuous at $x = 0,$ but can be made continuous at $x = 0$
  • D
    none of these.
✓

Answer

Correct option: B.
is not continuous at $x = 0$
Given, $\text{f(x)=}\begin{cases}\frac{\text{e}\frac{1}{\text{x}}-1}{\text{e}\frac{1}{\text{x}}+1},\text{x}\neq0\\0,\text{x}=0\end{cases}$
We have
$\lim\limits_{\text{x}\rightarrow0}\text{f(x)}=\lim\limits_{\text{x}\rightarrow0}\Bigg(\frac{\text{e}\frac{1}{\text{x}}-1}{\text{e}\frac{1}{\text{x}}+1}\Bigg)$
if $\text{e}^\frac{1}{\text{x}}=\text{t},$ then
$\text{x}\rightarrow0, \text{t}\rightarrow\infty$
$\lim\limits_{\text{x}\rightarrow0}\text{f(x)}=\lim\limits_{\text{t}\rightarrow\infty}\Big(\frac{\text{t}-1}{\text{t}+1}\Big)$
$=\lim\limits_{\text{t}\rightarrow\infty}\Bigg(\frac{1-\frac{1}{\text{t}}}{1+\frac{1}{\text{t}}}\Bigg)=\frac{1-0}{1+0}=1$
Also, $f(0) = 0$
$\because\ \lim\limits_{\text{x}\rightarrow0}\text{f(x)}\neq\text{f}(0)$
Hence, $f(x)$ is discontinuons at $x = 0.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Continuity→Continuity — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\cos^{-1}\frac{\text{x}}{3}+\cos^{-1}\frac{\text{y}}{2}=\frac{\theta}{2},$ then, $4\text{x}^2-12\text{xy}\cos^2\frac{\theta}{2}+9\text{y}^2=$
→
The position vectors of the points $A, B, C$ are $2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\ 3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{i}}+4\hat{\text{j}}-3\hat{\text{k}}$ respectively. These points,
→
The area bounded by $y= e ^x, \quad y= e ^{-x}$ and the straight line $x=1$ is
→
$\int \frac{\sqrt{\tan x}}{\sin x \cos x} dx =$____
→
What should be added in vector $\overline{a}=3 \hat{i}+4 \hat{j}-2 \hat{k}$ to get its resultant a unit vector $\hat{i}$ ?
→
If $\text{f(x)}=\frac{1}{1-\text{x}},$ then the set of points discontinuity of the function $f(f(f(x)))$ is:
→
The direction cosines of normal to the plane $\bar{r} .(2 \hat{i}-3 \hat{j}+\hat{k})+9=0$
→
If $P_1$ and $P_2$ are the lengths of the perpendiculars from the points $(2,3,4)$ and $(1,1,4)$ respectively from the plane $3 x-6 y+2 z+11=0$, then $P _1$ and $P _2$ are the roots of the equation
→
If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+\lambda \hat{j}+\hat{k}, \vec{c}=\hat{i}-\hat{j}+4 \hat{k}$ and $\vec{a} \cdot(\vec{b} \times \vec{c})=10$, then $\lambda$ is equal to
→
If $\sin ^{-1} \frac{4}{5}+\cos ^{-1} \frac{12}{13}=\sin ^{-1} \alpha$, then $\alpha=$
→