Differential and Integral Calculus by Feliciano and Uy, Exercise 1.3, Problem 14

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PROBLEM:

Evaluate lim⁡x→π4(tan⁡ 2xsec⁡ 2x)\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=π4x=\frac{\pi }{4} leads to the indeterminate form 00\frac{0}{0} which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows

lim⁡x→ π 4(tan⁡ 2xsec⁡ 2x)=lim⁡x→π4(sin⁡ 2xcos⁡ 2x1cos⁡ 2x)=lim⁡x→ π 4(sin⁡ 2x)=sin⁡(2⋅π4)=sin⁡π2=1  (Answer)\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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