Limit of a Function in Indeterminate Form| Differential and Integral Calculus| Feliciano and Uy| Exercise 1.3, Problem 14|

Evaluate

\lim\limits_{x\to \frac{\pi }{4}}\left(\frac{tan\:2x}{sec\:2x}\right)

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

\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

 

 

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