JEE Mains · Maths · STD 11 - 12. limits
If \(\alpha=\lim _{x \rightarrow 0^{+}}\left(\frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)\) and \(\beta=\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}\) are the roots of the quadratic equation \(a x^2+b x-\sqrt{e}=0\), then 12 \(\log _e(a+b)\) is equal to .............
- A \(4\)
- B \(6\)
- C \(5\)
- D \(1\)
Answer & Solution
Correct Answer
(B) \(6\)
Step-by-step Solution
Detailed explanation
\( \alpha=\lim _{x \rightarrow 0^{+}} e^{\sqrt{x}} \frac{\left(e^{\sqrt{\tan x}-\sqrt{x}}-1\right)}{\sqrt{\tan x}-\sqrt{x}} \) \( =1 \) \( \beta=\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x} \) \( =e^{1 / 2} \) \( x^2-(1+\sqrt{e})+\sqrt{e}=0 \) \( a x^2+b x-\sqrt{e}=0\)…
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