Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}\text{|x|}\cos\Big(\frac{1}{\text{x}}\Big), & \text{ x}\neq 0\\0 &\text{ x} = 0\end{cases}\text{at x}=0$
✓

Answer

Given,
$\text{f}\text{(x)}=\text{x}\cos\Big(\frac{1}{\text{x}}\Big),\text{x}\neq0$
$\text{f}\text{(x)}=0,\ \text{x}=0$
We observe
$\lim\limits_{\text{x} \rightarrow 0}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow 0}\text{x}\cos\Big(\frac{1}{\text{x}}\Big)$
$\lim\limits_{\text{x} \rightarrow 0}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow 0}\text{x}\lim\limits_{\text{x} \rightarrow 0}\cos\Big(\frac{1}{\text{x}}\Big)$
$\lim\limits_{\text{x} \rightarrow 0}\text{f}\text{(x)}=0\times\lim\limits_{\text{x} \rightarrow 0}\cos\Big(\frac{1}{\text{x}}\Big)$
$=0$
$\lim\limits_{\text{x} \rightarrow 0}\text{f}\text{(x)}=\text{f}(0)$
Hence, f(x) is continuous 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 "3 Marks Question" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\text{A}=\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix},$ then show that $\text{A}^2=\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix}.$
→
If the position vector $\vec{\text{a}}$ of a point (12, n) is such that $\big|\vec{\text{a}}\big|=13$, find the value (s) of n.
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\log\tan\text{x dx}$
→
Evalute the following integrals:
$\int\sqrt{\frac{1-\sin2\text{x}}{1+\sin2\text{x}}}\text{dx}$
→
Evaluate the following integrals:
$\int\frac{\sin2\text{x}}{(\text{a}+\text{b}\cos2\text{x})^2}\text{dx}$
→
If a vector$​​\vec{\text{a}}$ is perpendicular to two non-collinear vectors $\vec{\text{b}}$ and $\vec{\text{c}},$ then show that $​​\vec{\text{a}}$ is perpendicular to every vector in the plane of $\vec{\text{b}}$ and $\vec{\text{c}}.$
→
If A is a $3 \times 3$ invertible matrix, then what will be the value of k if det $(A^{-1}) = (det A)^k$.
→
Find X if $\text {Y} = \begin{bmatrix}3&2\\1&4\end{bmatrix} \text{and}\ \text{2X + Y} = \begin{bmatrix}1&0\\-3&2\end{bmatrix}.$
→
If A = {1, 2, 3, 4} define relations on A which have properties of being:
Reflexive, transitive but not symmetric.
→
If a machine is correctly set up it produces $90 \%$ acceptable items. If it is incorrectly set up it produces only $40 \%$ acceptable item. Past experience shows that $80 \%$ of the setups are correctly done. If after a certain set up, the machine produces $2$ acceptable items, find the probability that the machine is correctly set up.
→