MHT CET · Maths · Limits
The value of \(\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}\) is
- A \(\frac{1}{3 \sqrt{3}}\)
- B \(\frac{2}{\sqrt{3}}\)
- C \(\frac{2}{3 \sqrt{3}}\)
- D \(\frac{4}{3 \sqrt{3}}\)
Answer & Solution
Correct Answer
(C) \(\frac{2}{3 \sqrt{3}}\)
Step-by-step Solution
Detailed explanation
\(\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}} \)
\( =\lim _{x \rightarrow a} \frac{(\sqrt{a+2 x}-\sqrt{3 x})(\sqrt{a+2 x}+\sqrt{3 x})(\sqrt{3 a+x}+2 \sqrt{x})}{(\sqrt{3 a+x}-2 \sqrt{x})(\sqrt{3 a+x}+2 \sqrt{x})(\sqrt{a+2 x}+\sqrt{3 x})} \)
\( =\lim _{x \rightarrow a} \frac{(a+2 x-3 x)(\sqrt{3 a+x}+2 \sqrt{x})}{(3 a+x-4 x)(\sqrt{a+2 x}+\sqrt{3 x})} \)
\( =\lim _{x \rightarrow a} \frac{(a-x)(\sqrt{3 a+x}+2 \sqrt{x})}{3(a-x)(\sqrt{a+2 x}+\sqrt{3 x})}\)
\( =\frac{1}{3} \cdot \frac{(\sqrt{3 a+a}+2 \sqrt{a})}{(\sqrt{a+2 a}+\sqrt{3 a})} \)
\( =\frac{4 \sqrt{a}}{3 \times 2 \times \sqrt{3} \cdot \sqrt{a}} \)
\(=\frac{2}{3 \sqrt{3}}\)
\( =\lim _{x \rightarrow a} \frac{(\sqrt{a+2 x}-\sqrt{3 x})(\sqrt{a+2 x}+\sqrt{3 x})(\sqrt{3 a+x}+2 \sqrt{x})}{(\sqrt{3 a+x}-2 \sqrt{x})(\sqrt{3 a+x}+2 \sqrt{x})(\sqrt{a+2 x}+\sqrt{3 x})} \)
\( =\lim _{x \rightarrow a} \frac{(a+2 x-3 x)(\sqrt{3 a+x}+2 \sqrt{x})}{(3 a+x-4 x)(\sqrt{a+2 x}+\sqrt{3 x})} \)
\( =\lim _{x \rightarrow a} \frac{(a-x)(\sqrt{3 a+x}+2 \sqrt{x})}{3(a-x)(\sqrt{a+2 x}+\sqrt{3 x})}\)
\( =\frac{1}{3} \cdot \frac{(\sqrt{3 a+a}+2 \sqrt{a})}{(\sqrt{a+2 a}+\sqrt{3 a})} \)
\( =\frac{4 \sqrt{a}}{3 \times 2 \times \sqrt{3} \cdot \sqrt{a}} \)
\(=\frac{2}{3 \sqrt{3}}\)
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