Download App

Continuity — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsContinuity5 Marks
Question
Show that $\text{f}\text{ (x)}=\begin{cases}\frac{\text{|x}-\text{a}|}{|\text{x}-\text{a}|}, & \text{when} \text{ x}\neq 0\\2, & \text{when}\text{ x} = 0\end{cases}$ is discontinuous at x = a.
✓

Answer

The given function can be rewritten as:
$\text{f}\text{(x)}=\begin{cases}\frac{\text{z}-\text{a}}{\text{z}-\text{a}}, & \text{when} \text{ x}> 0\\\frac{\text{a}-\text{x}}{\text{z}-\text{a}}, & \text{when}\text{ x} < 0\\ 1,&\text{when}\text{ x} = \text{a}\end{cases}$
$\text{f}\text{(x)}=\begin{cases}1, & \text{when} \text{ x}> \text{a}\\-1, & \text{when}\text{ x} < \text{a}\\ 1,&\text{when x}= \text{a}\end{cases}$
$\text{f}\text{(x)}=\begin{cases}1, & \text{when} \text{ x}\geq \text{a}\\-1, & \text{when}\text{ x} < \text{a}\end{cases}$
We observe
$(\text{LHL at x}=\text{a})=\lim\limits_{\text{z} \rightarrow \text{a}^-}\text{f}\text{(x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}\text{(a}-\text{h})$
$=\lim\limits_{\text{h} \rightarrow 0}(-1)=-1$
$(\text{RHL at x}=\text{a})=\lim\limits_{\text{x} \rightarrow \text{a}^+}\text{f}\text{(x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}\text{(a}+\text{h})$
$=\lim\limits_{\text{h} \rightarrow 0}(1)=1$
$\therefore\lim\limits_{\text{x} \rightarrow \text{a}^-}\text{f}\text{(x)}\neq\lim\limits_{\text{x} \rightarrow \text{a}^+}\text{f}\text{(x)}$
Thus, f(x) is discontinuous at x = a

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

Evaluate the following integrals:$\int\limits^4_1\big\{|\text{x}-1|+|\text{x}-2|+|\text{x}-4\big\}\text{dx}$
→
Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(\hat{\text{i}}+\hat{\text{j}}\big)+\lambda\big(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}+\mu\big(3\hat{\text{i}}-5\hat{\text{j}}+2\hat{\text{k}}\big)$
→
A population grows at the rate of $5$% per year. How long does it take for the population to double?
→
Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\frac{\text{dy}}{\text{dx}}\cos^2\text{x}=\tan\text{x}-\text{y}$
→
Evaluate the following intregals: $\int\frac{1}{\sqrt{3}\sin\text{x}+\cos\text{x}}\ \text{dx}$
→
Find the vector equations of the coordinate planes.
→
The height of a cone increases by k%, its semi-vertical angle remaining the same. What is the approximate percentage increase
  1. In total surface area, and
  2. In the volume, assuming that k is small?
→
 A particle moves along the curve $y=x^2+2 x$. At what point(s) on the curve are the $x$ and $y$ coordinates of the particle changing at the same rate?
→
A(1, 0, 4) B(0, -11, 1), C(2, -3, 1) are three points and D is the fool of perpendicular from A on BC. Find the coordinates of D.
→
Let X denot the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
$\text{P}(\text{X = x})=\begin{cases}\text{kx},&\text{if}\text{ x}=0\text{ or }1\\2\text{kx},&\text{if x = 2}\\\text{k}(5-\text{x}),&\text{if x = 3 or 4}\\0,&\text{if x > 4}\end{cases}$
where k is a positive constant. Find the value of k. Also find the probability that you will get addmission in
  1. Exactly one college.
  2. At most two colleges.
  3. At least two colleges.
→