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  4. Evaluate displaystyle lim x arrow ∞ 2x-1 tan ((a/2x)).

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Q. Evaluate $\displaystyle\lim _{x \rightarrow \infty} 2^{x-1} \tan \left(\frac{a}{2^{x}}\right)$.

Limits and Derivatives

Solution:

We have
$\displaystyle\lim _{x \rightarrow \infty} 2^{x-1} \tan \left(\frac{a}{2^{x}}\right)=\displaystyle
\lim _{x \rightarrow \infty} \frac{a}{2} \frac{\tan \left(\frac{a}{2^{x}}\right)}{\left(\frac{a}{2^{x}}\right)}\left(\frac{0}{0}\right.$ form $)$
$=\frac{a}{2} \lim _{y \rightarrow 0} \frac{\tan y}{y}=\frac{a}{2}$
(where $y=\frac{a}{2^{x}}$ )