TS EAMCET · Maths · Limits
\(\lim _{x \rightarrow 0} \frac{\left(3^{2 x}-\sqrt{x+1}\right) \sin 5 x}{1-\cos 4 x}=\)
- A \(\frac{3}{5}(\log 18-1)\)
- B \(\frac{5}{16} \log \left(\frac{81}{e}\right)\)
- C \(\frac{4}{15}(\log 81-1)\)
- D \(\frac{16}{5}[\log (27)-1]\)
Answer & Solution
Correct Answer
(B) \(\frac{5}{16} \log \left(\frac{81}{e}\right)\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text \lim _{x \rightarrow 0} \frac{\left(3^{2 x}-\sqrt{x+1}\right)}{1-\cos 4 x} \sin 5 x \\ & =\lim _{x \rightarrow 0} \frac{\sin 5 x}{2 \sin ^2 2 x} \times \frac{\left(3^{2 x}-\sqrt{x+1}\right) \times\left(3^{2 x}+\sqrt{x+1}\right)}{3^{2 x}+\sqrt{x+1}} \\ &…
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