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PROBLEM:
Evaluate \displaystyle \lim\limits_{x\to \frac{\pi }{4}}\left(\frac{\tan\:2x}{\sec\:2x}\right).
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SOLUTION:
A straight substitution of x=\frac{\pi }{4} leads to the indeterminate form \frac{0}{0} which is meaningless.
Therefore, to evaluate the limit of the given function, we proceed as follows
\begin{align*}
\displaystyle \lim\limits_{x\to \:\frac{\pi \:}{4}}\left(\frac{\tan\:2x}{\sec\:2x}\right) & =\lim\limits_{x\to \frac{\pi }{4}}\left(\frac{\frac{\sin\:2x}{\cos\:2x}}{\frac{1}{\cos\:2x}}\right) \\
\\
& =\lim\limits_{x\to \:\frac{\pi \:}{4}}\left(\sin\:2x\right) \\
\\
& =\sin\left(2\cdot \frac{\pi }{4}\right) \\
\\
& =\sin\frac{\pi }{2} \\
\\
& =1 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}
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