Search any question & find its solution
Question:
Answered & Verified by Expert
$\lim _{x \rightarrow a} \frac{\sqrt{3 x-a}-\sqrt{x+a}}{x-a}=$
Options:
Solution:
1226 Upvotes
Verified Answer
The correct answer is:
$1 / \sqrt{2 a}$
$\begin{aligned}
& \lim _{x \rightarrow a} \frac{\sqrt{3 x-a}-\sqrt{x+a}}{x-a} \\
& \quad=\lim _{x \rightarrow a} \frac{\sqrt{3 x-a}-\sqrt{x+a}}{(x-a)} \times \frac{\sqrt{3 x-a}+\sqrt{x+a}}{\sqrt{3 x-a}+\sqrt{x+a}} \\
& \quad=\frac{2}{2 \sqrt{2 a}}=\frac{1}{\sqrt{2 a}}
\end{aligned}$
Aliter : Apply L-Hospital's rule
$\begin{aligned}
\lim _{x \rightarrow a} \frac{\sqrt{3 x-a}-\sqrt{x+a}}{x-a} & =\lim _{x \rightarrow a} \frac{3}{2 \sqrt{3 x-a}}-\frac{1}{2 \sqrt{x+a}} \\
& =\frac{3}{2 \sqrt{2 a}}-\frac{1}{2 \sqrt{2 a}}=\frac{1}{\sqrt{2 a}} .
\end{aligned}$
& \lim _{x \rightarrow a} \frac{\sqrt{3 x-a}-\sqrt{x+a}}{x-a} \\
& \quad=\lim _{x \rightarrow a} \frac{\sqrt{3 x-a}-\sqrt{x+a}}{(x-a)} \times \frac{\sqrt{3 x-a}+\sqrt{x+a}}{\sqrt{3 x-a}+\sqrt{x+a}} \\
& \quad=\frac{2}{2 \sqrt{2 a}}=\frac{1}{\sqrt{2 a}}
\end{aligned}$
Aliter : Apply L-Hospital's rule
$\begin{aligned}
\lim _{x \rightarrow a} \frac{\sqrt{3 x-a}-\sqrt{x+a}}{x-a} & =\lim _{x \rightarrow a} \frac{3}{2 \sqrt{3 x-a}}-\frac{1}{2 \sqrt{x+a}} \\
& =\frac{3}{2 \sqrt{2 a}}-\frac{1}{2 \sqrt{2 a}}=\frac{1}{\sqrt{2 a}} .
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.