Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Find which of the function:
$\text{f(x)}=\begin{cases}|\text{x}|\cos\frac{1}{\text{x}},&\text{if x}\neq0\\0,&\text{if x}=0\end{cases}$
at x = 0
✓

Answer

We have, $\text{f(x)}=\begin{cases}|\text{x}|\cos\frac{1}{\text{x}},&\text{if x}\neq0\\0,&\text{if x}=0\end{cases}$ at x = 0
At x = 0, $\text{L.H.L}=\lim\limits_{\text{h}\rightarrow0}|\text{x}|\cos\frac{1}{\text{x}}$
$=\lim\limits_{\text{h}\rightarrow0}|0-\text{h}|\cos\frac{1}{0-\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\cos\frac{1}{\text{h}}$
= 0 × [an oscillating number between -1 and 1] = 0
$\text{R.H.L}=\lim\limits_{\text{x}\rightarrow0^+}|\text{x}|\cos\frac{1}{\text{x}}$
$=\lim\limits_{\text{h}\rightarrow0}|0+\text{h}|\cos\frac{1}{0+\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\cos\frac{1}{\text{h}}$
= 0 × [an oscillating number between -1 and 1] = 0
Also f(0) = 0 (given)
Thus, L.H.L = R.H.L = f(0)
Hence, f(x) is discontinuous 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 "5 Marks Questions" 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

A bag contains 3 red and 5 black balls and a second bag contains 6 red and 4 black balls. A ball is drawn from each bag. Find the probability that one is red and the other is black.
→
Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}0&2&6\\1&5&0\\3&7&1 \end{vmatrix}$
→
Evaluate the following integrals:$\int\text{x}\tan^2\text{x dx}$
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = (x - 1)(x - 2)^2$
→
If $\vec{\text{p}}=5\hat{\text{i}}+\lambda\hat{\text{j}}-3\hat{\text{k}}$ and $\vec{\text{q}}=\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}},$ then find the value of $\lambda,$ so that $\vec{\text{p}}+\vec{\text{q}}$ and $\vec{\text{p}}-\vec{\text{q}}$ are perpendicular vectora.
→
Evaluvate the following intregals:
$\int\frac{4\sin\text{x}+5\cos\text{x}}{5\sin\text{x}+4\cos\text{x}}\ \text{dx}$
→
Differentiate the following functions with respect to x:
$\text{e}^{\text{ax}}\sec\text{x}\tan2\text{x}$
→
$\begin{vmatrix}\text{b}+\text{c}&\text{a}&\text{a}\\\text{b}&\text{c}+\text{a}&\text{b}\\\text{c}&\text{c}&\text{a}+\text{b}\end{vmatrix}=4\text{abc}$
→
$\int\frac{\text{x}}{\sqrt{\text{x}+\text{a}}-\sqrt{\text{x}+\text{b}}}\text{dx}$
→
A small firm manufacturers items $A$ and $B.$ The total number of items $A$ and $B$ that it can manufacture in a day is at the most $24.$ Item A takes one hour to make while item $B$ takes only half an hour. The maximum time available per day is $16$ hours. If the profit on one unit of item $A$ be Rs. $300$ and one unit of item $B$ be Rs. $160,$ how many of each type of item be produced to maximize the profit? Solve the problem graphically.
→