Download App

Continuity — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity5 Marks
Question
If the functions f(x), defined below is continuous at x = 0, find the value of k.
$\text{f(x)}=\begin{cases}\frac{1-\cos2\text{x}}{2\text{x}^2},&\text{x}<0\\\text{k},&\text{x}=0\\\frac{\text{x}}{|\text{x}|},&\text{x}>0\end{cases}$
✓

Answer

Given, $\text{f(x)}=\begin{cases}\frac{1-\cos2\text{x}}{2\text{x}^2},&\text{x}<0\\\text{k},&\text{x}=0\\\frac{\text{x}}{|\text{x}|},&\text{x}>0\end{cases}$
$\text{f(x)}=\begin{cases}\frac{1-\cos2\text{x}}{2\text{x}^2},&\text{x}<0\\\text{k},&\text{x}=0\\1,&\text{x}>0\end{cases}$
We have,
$(\text{LHL at x}= 0)=\lim_\limits{\text{x}\rightarrow0^-}\text{f(x)}=\lim_\limits{\text{h}\rightarrow0}\text{f}(0-\text{h})$
$=\lim_\limits{\text{h}\rightarrow 0}\Big(\frac{1-\cos2(-\text{h})}{2(-\text{h})^2}\Big)$
$=\lim_\limits{\text{h}\rightarrow 0}\Big(\frac{1-\cos2\text{h}}{2\text{h}^2}\Big)$
$=\frac{1}{2}\lim_\limits{\text{h}\rightarrow 0}\Big(\frac{2\sin^2\text{h}}{\text{h}^2}\Big)$
$=\frac{2}{2}\lim_\limits{\text{h}\rightarrow 0}\Big(\frac{\sin^2\text{h}}{\text{h}^2}\Big)$
$=\frac{2}{2}\lim_\limits{\text{h}\rightarrow 0}\Big(\frac{\sin^2\text{h}}{\text{h}}\Big)^2$
$=1\times1$
$=1$
$(\text{RHL at x}= 0)=\lim_\limits{\text{x}\rightarrow0^+}\text{f(x)}=\lim_\limits{\text{h}\rightarrow0}\text{f}(0+\text{h})$
$=\lim_\limits{\text{h}\rightarrow 0}\text{f(h)}=\lim_\limits{\text{h}\rightarrow 0}(1)=1$
Also, $\text{f}(0)=\text{k}$
If f(x) is continuous at x = 0, then
$\lim_\limits{\text{x}\rightarrow 0^-}\text{f(x)}=\lim_\limits{\text{x}\rightarrow 0^+}=\text{f}(0)$
$\Rightarrow1=1=\text{k}$
Hence, the required value of k is 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 "5 Marks Questions" from Continuity→Continuity — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{ae}^{\theta}(\sin\theta-\cos\theta),\text{y}=\text{ae}^\theta(\sin\theta+\cos\theta)$
→
Solve the following systems of linear equations by cramer's rule:
x + 2y = 1,
3x + y = 4
→
Using differentials, find the approximate values of the following:
$\sqrt{0.082}$
→
Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}(\text{x}+2\text{y})}{\text{x}(2\text{x}+\text{y})},\text{y}(1)=2$
→
Find the angle between the lines whose direction cosines are given by the equations:
$2l + 2m - n = 0, mn + ln + lm = 0$
→
If $\text{e}^{\text{x}}+\text{e}^{\text{y}}=\text{e}^{\text{x}+\text{y}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{e}^{\text{x}}(\text{e}^\text{y}-1)}{\text{e}^{\text{y}}(\text{e}^{\text{x}}-1)}$ or $\frac{\text{dy}}{\text{dx}}+\text{e}^{\text{y}-\text{x}}=0$
→
Of all the closed cylindrical cans (right circular), which enclose a given volume of $100cm^3,$ which has the minimum surface area ?
→
Form the differential equation of the family of curves represented by the equation (a being the perimeter):$(2\text{x}-\text{a})^2-\text{y}^2=\text{a}^2$
→
Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.
→
A rubber company is engaged in producing three types of tyres A, B and C. Each type requires processing in two plants, Plant I and Plant II. The capacities of the two plants, in number of tyres per day, are as follows:
Plant
A
B
C
I
50
100
100
II
60
60
200
The monthly demand for tyre A, B and C is 2500, 3000 and 7000 respectively. If plant I costs Rs. 2500 per day, and plant II costs Rs. 3500 per day to operate, how many days should each be run per month to minimize cost while meeting the demand? Formulate the problem as LPP.
→