Download App

Continuity — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity5 Marks
Question
Determine if $\text{f(x)}=\begin{cases}\text{x}^2\sin\frac{1}{\text{x}},&\text{ x}\neq0\\0,&\text{x}=0\end{cases}$ is a continuous function?
✓

Answer

The given function f is $\text{f(x)}=\begin{cases}\text{x}^2\sin\frac{1}{\text{x}},&\text{ x}\neq0\\0,&\text{x}=0\end{cases}$
It is evident that f is defind at all points of the real line.
Let c be a real number.
Case I:
If $\text{c}\neq0,$ then $\text{f(c)}=\text{c}^2\sin\frac{1}{\text{c}}$
$\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{f(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}\Big(\text{x}^2\sin\frac{1}{\text{x}}\Big)=\Big(\lim\limits_{{\text{x}}\rightarrow\text{c}}\sin\frac{1}{\text{x}}\Big)=\text{c}^2\sin\frac{1}{\text{c}}$
$\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{f(x)}=\text{f(c)}$
So, f is continuouse at all points $\text{x}\neq0$
Case II:
If c = 0, then f(0) = 0
$\lim\limits_{{\text{x}}\rightarrow0^-}\text{f(x)}=\lim\limits_{{\text{x}}\rightarrow0^-}\Big(\text{x}^2\sin\frac{1}{\text{x}}\Big)=\lim\limits_{{\text{x}}\rightarrow0}\Big(\text{x}^2\sin\frac{1}{\text{x}}\Big)$
It is know that $-1\leq\sin\frac{1}{\text{x}}\leq1,\text{x}\neq0$
$\Rightarrow-\text{x}^2\leq\text{x}^2\sin\frac{1}{\text{x}}\leq\text{x}^2$
$\Rightarrow\lim\limits_{{\text{x}}\rightarrow0}(-\text{x}^2)\leq\lim\limits_{{\text{x}}\rightarrow0}\Big(\text{x}^2\sin\frac{1}{\text{x}}\Big)\leq\lim\limits_{{\text{x}}\rightarrow0}\text{x}^2$
$\Rightarrow0\leq\lim\limits_{{\text{x}}\rightarrow0}\Big(\text{x}^2\sin\frac{1}{\text{x}}\Big)\leq0$
$\Rightarrow\lim\limits_{{\text{x}}\rightarrow0}\Big(\text{x}^2\sin\frac{1}{\text{x}}\Big)=0$
$\Rightarrow\lim\limits_{{\text{x}}\rightarrow0^-}\text{f(x)}=0$
Similarly, $\lim\limits_{{\text{x}}\rightarrow0^+}\Big(\text{x}^2\sin\frac{1}{\text{x}}\Big)=\lim\limits_{{\text{x}}\rightarrow0}\Big(\text{x}^2\sin\frac{1}{\text{x}}\Big)=0$
$\therefore\ \lim\limits_{{\text{x}}\rightarrow0^-}\text{f(x)}=\text{f}(0)=\lim\limits_{{\text{x}}\rightarrow0^-}\text{f(x)}$
So, f is continuous at x = 0
From the above observations, it can be continuouse that f is continuous at every point of the real line.
Thus, f is a continuous function.

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 "5 Marks Questions" from Continuity→Continuity — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following integrals as limit of sum:
$\int\limits^3_1(3\text{x}-2)\text{dx}$
→
Show that $\text{y}=\frac{\text{c}-\text{x}}{1+\text{cx}}$ is a solution of the differential equation $(1+\text{x}^2)\frac{\text{dy}}{\text{dx}}+(1+\text{y}^2)=0.$
→
Solve the following determinant equations:
$\begin{vmatrix}\text{x}+\text{a}&\text{x}&\text{x}\\\text{x}&\text{x}+\text{a}&\text{x}\\\text{x}&\text{x}&\text{x}+\text{a}\end{vmatrix}=0,\text{a}\neq0$
→
Evaluate the following integrals:
$\int^\limits\frac{\pi}{2}_{0}\frac{\cos^2\text{x}}{1+3\sin^3\text{x}}\text{ dx}$
→
If $\overrightarrow{\text{a}}\text{ and } \overrightarrow{\text{b}}$are two vectors such that $|\overrightarrow{\text{a}} + \overrightarrow{\text{b}}| = | \overrightarrow{\text{a}}|,$ then prove that vector 2 $\overrightarrow{\text{a}} + \overrightarrow{\text{b}}$is perpendicular to vector$\overrightarrow{\text{b}}.$
→
Show that the function $\text{f}(\text{x})=\sin\Big(2\text{x}+\frac{\pi}{4}\Big)$ is decreasing on $\Big(\frac{3\pi}{8},\frac{5\pi}{8}\Big).$
→
Evalute the following integrals:
$\int\frac{\cos4\text{x}-\cos2\text{x}}{\sin4\text{x}-\sin2\text{x}}\text{dx}$
→
Find the radius of the circular section of the sphere $|\vec{\text{r}}|=5$ by the plane $\vec{\text{r}}.(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})=3\sqrt{3}.$
→
Evaluate the following integrals:
$\int\limits^{\pi}_0\frac{\text{x}\sin\text{x}}{1+\sin\text{x}}\text{ dx}$
→
In a game, a man wins ₹ 5 for getting a number greater than 4 and loses ₹ 1 otherwise, when a fair die is thrown. The man decided to throw a die three but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses.
→