Download App

Continuity — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsContinuity5 Marks
Question
Prove that the function
$\text{f}\text{(x)}=\begin{cases}\frac{\text{x}}{|\text{x|+2}\text{x}^2}, &\text{ x}\neq0\\\text{k}, &\text{ x}=0\end{cases}$
remains discontinuous at x = 0, regardless the choice of k.
✓

Answer

The given function can be rewritten as,
$\text{f}\text{(x)}=\begin{cases}\frac{\text{x}}{\text{x}+2\text{x}^2}, & \text{x} > 0\\\frac{-\text{x}}{\text{x}-2\text{x}^2}, &\text{x} <0\\ \text{k},&\text{x}=0\end{cases}$
$\Rightarrow\text{f}\text{(x)}=\begin{cases}\frac{1}{2\text{x}+1}, &\text{x} > 0\\\frac{1}{2\text{x}-1}, & \text{x} <0\\\text{k},&\text{x} =0\end{cases}$
We observe
$\text{(LHL at x}=0)\lim\limits_{\text{x} \rightarrow 0^-}\text{f}\text{(x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}(0-\text{h)}$
$=\lim\limits_{\text{h} \rightarrow 0}\text{f}(-\text{h)}=\lim\limits_{\text{h} \rightarrow 0}\frac{1}{-2\text{h}-1}=-1$
$\text{(RHL at x}=0)\lim\limits_{\text{x} \rightarrow 0^+}\text{f}\text{(x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}(0+\text{h)}$
$=\lim\limits_{\text{h} \rightarrow 0}\text{f}(\text{h)}=\lim\limits_{\text{h} \rightarrow 0}\frac{1}{2\text{h}+1}=1$
So,$=\lim\limits_{\text{x} \rightarrow 0^-}\text{f}\text{(x)}\neq\lim\limits_{\text{x} \rightarrow 0^+}\text{f}\text{(x)}$ Such that
$=\lim\limits_{\text{x} \rightarrow 0^-}\text{f}\text{(x)}\&\lim\limits_{\text{x} \rightarrow 0^+}\text{f}\text{(x)}$
are independent of k.
Thus, f(x) is discontinuous at x = 0,
regardless of the choice of k.

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 "Solve the Following Question.(5 Marks)" from Continuity→Continuity — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

find the area of the circle $x^2 + y^2 = 16$ which is exterior to the parabola $y^2 = 6x.$
→
Two dice are drawn together and the number appearing on them noted. X denotes the sum of the two numbers. Assuming that the 36 outcomes are equally likely, what is the probability distribution of X?
→
Solve the following determinant equations:
$\begin{vmatrix}1&\text{x}&\text{x}^3\\1&\text{b}&\text{b}^3\\1&\text{c}&\text{c}^3\end{vmatrix}=0,\text{b}\neq\text{c}$
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}+\frac{1+\text{y}^2}{\text{y}}=0$
→
A beam is supported at the two ends and is uniformly loaded. The bending moment M at a distance x from one end is given by
$\text{M}=\frac{\text{WL}}{2}\text{x}-\frac{\text{W}}{3}\frac{\text{x}^{3}}{\text{L}^{2}}$
Find the point at which M is maximum in each case.
→
A manufacturer makes two products A and B. Product A sells at Rs. 200 each and takes 1/2 hour to make. Product B sells at Rs. 300 each and takes 1 hour to make. There is a permanent order for 14 of product A and 16 of product B. A working week consists of 40 hours of production and weekly turnover must not be less than Rs 10000. If the profit on each of product A is Rs. 20 and on product B is Rs. 30, then how many of each should be produced so that the profit is maximum. Also, find the maximum profit.
→
Solve the following systems of homogeneous linear equations by matrix method:
$x + y + z = 0$
$x - y - 5z = 0$
$x + 2y + 4z = 0$
→
Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}(\text{x})=(\text{x}+1)(\text{x}+2)^\frac{1}{3}$
→
Find the equation of a plane which is at a distance of $3\sqrt{3}\text{ units}$ from the origin and the normal to which is equally inclined to the coordinate axes.
→
Verify Rolle's theorem for the following function on the indicated intervals $f(x) = x(x^- 2)^2$​​​​​​​ on the interval $[0, 2]$
→