AP EAMCET · Maths · Limits
If \(\alpha=\lim _{x \rightarrow 0} \frac{x \cdot 2^x-x}{1-\cos x}\) and \(\beta=\lim _{x \rightarrow 0} \frac{x \cdot 2^x-x}{\sqrt{1+x^2}-\sqrt{1-x^2}}\), then
- A \(\alpha=\beta\)
- B \(2\alpha=\beta\)
- C \(\alpha=2 \beta\)
- D \(\alpha=3\beta\)
Answer & Solution
Correct Answer
(C) \(\alpha=2 \beta\)
Step-by-step Solution
Detailed explanation
\begin{aligned} \alpha & =\lim _{x \rightarrow 0} \frac{x \cdot 2^x-x}{1-\cos x} \\ \alpha & =\lim _{x \rightarrow 0} \frac{x\left(2^x-1\right)}{1-\cos x}=\lim _{x \rightarrow 0} \frac{x\left(2^x-1\right)}{2 \sin ^2 \frac{x}{2}} \\ & =\lim _{x \rightarrow 0} \frac{1}{2} \cdot…
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