JEE Mains · Maths · STD 11 - 12. limits
If \(\lim _{\mathrm{t} \rightarrow 0}\left(\int_0^1(3 x+5)^{\mathrm{t}} \mathrm{d} x\right)^{\frac{1}{t}}=\frac{\alpha}{5 \mathrm{e}}\left(\frac{8}{5}\right)^{\frac{2}{3}}\), then \(\alpha\) is equal to ________
- A 60
- B 62
- C 64
- D 66
Answer & Solution
Correct Answer
(C) 64
Step-by-step Solution
Detailed explanation
\begin{aligned} \frac{\alpha}{5 e} & =\exp \left(\lim _{t \rightarrow 0} \frac{1}{t}\left(\int_0^1(3 x+5)^t d x-1\right)\right) \\ & =\exp \left(\lim _{t \rightarrow 0} \frac{1}{t}\left(\frac{(3 x+5)^{t+1}}{3(t+1)}\right)_0^1-1\right) \\ & =\exp \left(\lim _{t \rightarrow 0}…
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