Download App

Continuity — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity3 Marks
Question
Discuss the continuity of the function f(x) at the point $\text{x}=\frac{1}{2}$ where
$\text{f}\text{(x)}=\begin{cases}\text{x}, & 0\leq\text{x} < \frac{1}{2}\\\frac{1}{2},&\text{x}=\frac{1}{2}\\1-\text{x}, &\frac{1}{2}< \text{x}\leq 1\end{cases}$
✓

Answer

We want to discuss the continuity of the function at $\text{x}=\frac{1}{2}.$
$\text{LHL}=\lim\limits_{\text{x} \rightarrow \frac{1}{2}^-}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow 0}\text{f}\Big(\frac{1}{2}-\text{h}\Big)=\lim\limits_{\text{x} \rightarrow 0}\frac{1}{2}-\text{h}=\frac{1}{2}$
$\text{RHL}=\lim\limits_{\text{x} \rightarrow \Big(\frac{1}{2}^-\Big)}=\lim\limits_{\text{x} \rightarrow 0}\text{f}\Big(\frac{1}{2}+\text{h}\Big)=\lim\limits_{\text{x} \rightarrow 0}1-\Big(\frac{1}{2}+\text{h}\Big)=\frac{1}{2}$
$\text{f}\Big(\frac{1}{2}\Big)=\frac{1}{2}$
Thus, $\text{LHL}=\text{RHL}=\text{f}\Big(\frac{1}{2}\Big)=\frac{1}{2}$
Hence, the function is continuous at $\text{x}=\frac{1}{2}.$

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 "3 Marks Question" 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 the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}(\text{x})=\frac{1}{\text{x}^{2}+2}$
→
Check the commutativity and associativity of the following binary operations:
'*' on N defined by a * b = gcd(a, b) for all a, b ∈ N.
→
For the following differntial equations verify that the accompanying function is a solution:
Differential equation Function
$\text{x}+\text{y}\frac{\text{dy}}{\text{dx}}=0$ $\text{y}=\pm\sqrt{\text{a}^2-\text{x}^2}$
→
Evalute the following integrals:
$\int\frac{1}{\sqrt{1+\cos\text{x}}}\text{dx}$
→
Evaluate the following integrals:
$\int\frac{\text{cosec }\text{x}}{\text{cosec }\text{x}-\cot\text{x}}\text{dx}$
→
Let $\vec{a}=i+4 j+2 k, b=3 i-2 j+7 k$ and $c=2 i-j+4 k$. Find a vector $d$ which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} . \vec{d}=15$.
→
Evaluate the definite integral in Exercise:
$\int_{0}^{2}\frac{6\text{x}+3}{\text{x}^{2}+4}\text{dx}$
→
Let A be the set of all human beings in a town at a particular time. Determine whether the following relations are reflexive, symmetric and transitive:
R = {(x, y): x and y work at the same place}
→
Test the continuity of the function on f(x) at the origin:
$\text{f}\ (\text{x})=\begin{cases}\frac{\text{x}}{\text{|x|}},& \text{x}\neq0\\1, & \text{x} = 0\end{cases}$
→
Find the particular solution of the differential equation $\left[x \sin ^2\left(\frac{y}{x}\right)-y\right] dx + x d y =0$, given that $y =\frac{\pi}{4}$ when $x =1$
→