Download App

Continuity and Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability1 Mark
Question
The value of $k(k<0)$ for which the function $f$ defined as $f(x)=\left\{\begin{array}{cc}\frac{1-\cos k x}{x \sin x} & , x \neq 0 \\ \frac{1}{2} & , x=0\end{array}\right.$ is continuous at $x=0$ is
✓

Answer

We have, $f(x)=\left\{\begin{array}{cc}\frac{1-\cos k x}{x \sin x}, & x \neq 0 \\ \frac{1}{2}, & x=0\end{array}\right.$
$\because f(x)$ is continuous at $x=0$.
$\therefore \lim _{x \rightarrow 0} \frac{1-\cos k x}{x \sin x}=\frac{1}{2} $
$\Rightarrow \lim _{x \rightarrow 0} \frac{2 \sin ^2 \frac{k x}{2}}{x^2 \frac{\sin x}{x}}=\frac{1}{2}$
$\Rightarrow \lim _{x \rightarrow 0} 2 \cdot \frac{k^2}{4}\left\{\frac{\sin \left(\frac{k x}{2}\right)}{\frac{k x}{2}}\right\}^2 \frac{1}{\frac{(\sin x)}{x}}=\frac{1}{2}$
$\Rightarrow \frac{k^2}{2}=\frac{1}{2} $
$\Rightarrow k= \pm 1$
But $k < 0 $
$\therefore k=-1$

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 "M.C.Q (1 Marks)" from Continuity and Differentiability→Continuity and Differentiability — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Conisder the matrices
$A=\left[\begin{array}{rrr} 2 & 1 & 3 \\ 3 & -2 & 1 \\ -1 & 0 & 1 \end{array}\right], B=\left[\begin{array}{rr} 1 & -2 \\ 2 & 1 \\ 4 & 3 \end{array}\right], C=\left[\begin{array}{lll} 1 & 2 & 6 \end{array}\right] $
Then, which of the following is not defined?
→
The area bounded by $y = x^2$ and $y = 1 - x^2$ is:
→
Vikas printing company takes fee of Rs. 28 to print a oversized poster and Rs. 7 for each colour of ink used. Raaj printing company does the same work and charges poster for Rs. 34 and Rs. 5.50 for each colour of ink used. If z is the colours of ink used, find the values of z such that Vikas printing company would charge more to print a poster than Raaj printing company.
→
Area of the region bounded by the curve $y^2 = 4x, y-$axis and the line $y = 3,$ is:
→
If $\text{A}=\begin{bmatrix}5&\text{x}\\\text{y}&0\end{bmatrix}$ and $A = A^T,$ then$:$
→
If $w$ is a non $-$ real cube root of unity and $n$ is not a multiple of $3,$ then $\begin{vmatrix}1&\omega^{\text{n}}&\omega^{2\text{n}}\\\omega^{2\text{n}}&1&\omega^{\text{n}}\\\omega^{\text{n}}&\omega^{2\text{n}}&1\end{vmatrix}$ is equal to:
→
Choose the correct answer from the given four options.The matrix $\text{P}=\begin{bmatrix}0&0&4\\0&4&0\\4&0&0\end{bmatrix}$ is a:
  1. Square matrix.
  2. Diagonal matrix.
  3. Unit matrix.
  4. None of these.
→
Solution of the differential equation $\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{d x-d y}{d x+d y}$, is
→
If $\text{A}=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 0 & 1 \\ \text{a} & \text{b} & 2 \end{bmatrix},$ then $aI + bA + 2 A^2$ equals:
→
If $\text{y}=\log_\text{e}\Big(\frac{\text{x}}{\text{a}+\text{bx}}\Big)^\text{x}$ then $\text{x}^3\text{y}_2=$
  1. $(\text{xy}_1-\text{y})^2$
  2. $(1+\text{y})^2$
  3. $\Big(\frac{\text{y}-\text{xy}_1}{\text{y}_1}\Big)^2$
  4. $\text{None of these}$
→