Download App

Continuity — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsContinuity2 Marks
Question
Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}\text{(x}-\text{a})\sin\Big(\frac{1}{\text{x}-\text{a}}\Big), & \text{x} \neq 0\\\ \ 0, & \text{x} = \text{a}\end{cases}\text{at x}=\text{a}$
✓

Answer

Given,
$\text{f}\text{(x)}=\begin{cases}\text{(x}-\text{a})\sin\Big(\frac{1}{\text{x}-\text{a}}\Big), & \text{x} \neq 0\\\ 0,\text{x} = \text{a}\end{cases}$
Putting x - a = y, we get
$\lim\limits_{\text{x} \rightarrow \text{a}}\text{(x}-\text{a})\sin\Big(\frac{1}{\text{x}-\text{a}}\Big)$
$=\lim\limits_{\text{y} \rightarrow 0}\text{y}\sin\Big(\frac{1}{\text{y}}\Big)$
$=\lim\limits_{\text{y} \rightarrow 0}\text{y}\lim\limits_{\text{y} \rightarrow 0}\sin\Big(\frac{1}{\text{y}}\Big)$
$=0 \times\lim\limits_{\text{y} \rightarrow 0}\sin\Big(\frac{1}{\text{y}}\Big)=0$
$\Rightarrow\lim\limits_{\text{x} \rightarrow \text{a}}\text{f}\text{(x)}=\text{f}\text{(a)}=0$
Hence, f(x) is continuous at x = a.

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 "Solve the Following Question.(2 Marks)" 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

The measure of acute angle between the lines whose direction ratios are 3, 2, 6 and $- 2 , 1 , 2$ is ...
→
Differentiate the following w.r.t. x :

$\frac{3}{5 \sqrt[3]{\left(2 x^2-7 x-5\right)^5}}$

→
Find the value of 'p' for which the vectors $3\hat{\text{i}}+2\hat{\text{j}}+9\hat{\text{k}}$ and $\hat{\text{i}}-2\text{p}\hat{\text{j}}+3\hat{\text{k}}$ are parallel.
→
Give example of matrices:
A and B such that AB = O but A ≠ 0, B ≠ 0.
→
$\sin^{-1}\frac{1}{2}-2\sin^{-1}\frac{1}{\sqrt2}$
→
If the line $y=4 x-5$ touches the curve $y^2=a x^3+b$ at the point $(2,3)$, find $a$ and $b$.
→
Find the co-ordinates of the point, which divides the line segment joining the points $A (2,-6,8)$ and $B (-1,3,-4)$ externally in the ratio $1: 3$.
→
If $\vec{\text{b}}$ is a unit vector such that $\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=8,$ find $|\vec{\text{a}}|.$
→
Prove the Theorem: (Section formula for external division) Let $\mathrm{A}(\bar{a})$ and $\mathrm{B}(\bar{b})$ be any two points in the space and $\mathrm{R}(\bar{r})$ be the third point on the line $\mathrm{AB}$ dividing the segment $\mathrm{AB}$ externally in the ratio $m: n$. Then $\bar{r}=\frac{m \bar{b}-n \bar{a}}{m-n}$.
→
Evaluate the following integrals:$\int\limits^{\text{e}^2}_\text{e}\frac{1}{\text{x}\log\text{x}}\text{ dx}$
→