Q53 / 120 · VITEEE 2019 · Mathematics
Evaluate $\lim _{x \rightarrow 2} \frac{\sqrt{(x+7)}-3 \sqrt{(2 x-3)}}{\sqrt[3]{(x+6)}-2 \sqrt[3]{(3 x-5)}}$.
- A$\frac{17}{9}$
- B$\frac{17}{18}$
- Correct answerC$\frac{34}{23}$
- D$\frac{26}{7}$
Solution
We have
$$
L=\lim _{x \rightarrow 2} \frac{\sqrt{(x+7)}-3 \sqrt{(2 x-3)}}{\sqrt[3]{(x+6)}-2 \sqrt[3]{(3 x-5)}}\left(\frac{0}{0} \text { form }\right)
$$
Let $x-2=t$ such that when $x \rightarrow 2, t \rightarrow 0$. Then
$$
\begin{array}{l}
L=\lim _{t \rightarrow 0} \frac{(t+9)^{\frac{1}{2}}-3(2 t+1)^{\frac{1}{2}}}{(t+8)^{\frac{1}{3}}-2(3 t+1)^{\frac{1}{3}}} \quad\left(\frac{0}{0} \text { form }\right) \\
=\frac{3}{2} \lim _{t \rightarrow 0} \frac{\left(1+\frac{t}{9}\right)^{\frac{1}{2}}-(2 t+1)^{\frac{1}{2}}}{\left(1+\frac{t}{8}\right)^{\frac{1}{3}}-(3 t+1)^{\frac{1}{3}}} \quad\left(\frac{0}{0} \text { form }\right) \\
=\frac{3}{2} \lim _{t \rightarrow 0} \frac{\frac{1}{2} \frac{t}{9}-(2 t) \frac{1}{2}-(3 t) \frac{1}{3}}{3}=\frac{3}{2} \frac{1}{\frac{18}{2}}-1=\frac{1}{24}-1 \quad 23
\end{array}
$$
Hence, the correct answer is option: 'C'.