Download App

Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability4 Marks
Question
If function $f(x)=\frac{\sqrt{x}-\sqrt{a}}{x-a}$, is continuous at $x=a$, then find value of $f(a)$.
✓

Answer

value of $\text{R.H.L.}$
$\lim _{h \rightarrow 0} f(a+0)=\lim _{h \rightarrow 0} \frac{a+h-a}{\sqrt{a+h}-\sqrt{a}}$
$=\lim _{h \rightarrow 0}\left[\frac{h}{\sqrt{a+h}-\sqrt{a}} \times \frac{\sqrt{a+h}+\sqrt{a}}{\sqrt{a+h}+\sqrt{a}}\right] $
$=\lim _{h \rightarrow 0} h \frac{(\sqrt{a+h}+\sqrt{a})}{a+h-a} $
$=\lim _{h \rightarrow 0} \frac{h(\sqrt{a+h}+\sqrt{a})}{h}=\sqrt{a}+\sqrt{a} $
$=2 \sqrt{a}$
$\because$function is continuous at $x=a$.
$ \begin{array}{rlrl} \therefore & f(a)  =\lim _{h \rightarrow 0} f(a+h) \\ & \therefore f(a) =2 \sqrt{a} \end{array} $

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 "Answer the question [ 4 mark question ]" from Continuity and Differentiability→Continuity and Differentiability — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

$\cos ^{-1} \frac{1-a^2}{1+a^2}+\cos ^{-1} \frac{1-b^2}{1+b^2}=2 \tan ^{-1} x$
→
Find the value of $\int_2^5 \frac{\sqrt{x}}{\sqrt{7-x+\sqrt{x}}} d x$
→
Prove that $\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} \frac{x}{y}\right)+\tan \left(\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} \frac{x}{y}\right)=\frac{2 y}{x}$
→
Find the value of $\int_0^\pi \frac{x \sin x}{1+\cos ^2 x} d x$
→
Solve the differential equation $\cos ^2 x \frac{d y}{d x}+y=\tan x\left(0 \leq x \leq \frac{\pi}{2}\right)$
→
If given function is continuous at $x=2$. Find the value of $k$ :$
f(x)=\left\{\begin{array}{cc}
\frac{x^3+x^2-16 x+20}{(x-2)^2}, & \text { when } x \neq 2 \\
k, & \text { when } x=2
\end{array}\right.
$
→
If $y=\sin \left[2 \tan ^{-1}\left(\sqrt{\frac{1-x}{1+x}}\right)\right]$ then find $\frac{d y}{d x}$.
→
Find the value of $\int_0^\pi \log (1+\cos x) d x$.
→
Find the value of $\int_0^\pi \frac{\sec x}{\sec x+\tan x} d x$.
→
Prove that $\int_a^b(a+b-x) d x=\int_a^b f(x) d x$ and using this find $\int_4^a \frac{f(x)}{f(x)+f(13-x)} d x$.
→