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  • Limit  <img alt="\mathrm{\lim _{x \rightarrow 1}\left(1-x^{2}\right) \tan \left(\frac{\pi x}{2}\right) ?}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7Bx%20%5C

Limit  \mathrm{\lim _{x \rightarrow 1}\left(1-x^{2}\right) \tan \left(\frac{\pi x}{2}\right) ?}

Option: 1

\frac{2}{\pi}


Option: 2

\frac{5}{\pi}


Option: 3

0


Option: 4

\frac{10}{\pi}


Answers (1)

best_answer

Decompose also \mathrm{1-x^{2}} :

\mathrm{\lim _{x \rightarrow 1}\left(1-x^{2}\right) \tan \frac{\pi x}{2}=\lim _{x \rightarrow 1} \frac{1-x}{\cos (\pi x / 2)}(1+x) \sin \frac{\pi x}{2}}

The second and third factors have limits 2 and 1 respectively; for the first fraction, consider instead

\mathrm{\lim _{x \rightarrow 1} \frac{\cos (\pi x / 2)}{x-1}=\lim _{x \rightarrow 1} \frac{\cos (\pi x / 2)-\cos (\pi / 2)}{x-1}}

which is the derivative at 1 of the function \mathrm{f(x)=\cos (\pi x / 2)}; since
\mathrm{f^{\prime}(x)=-\frac{\pi}{2} \sin \frac{\pi x}{2}}

and so \mathrm{f^{\prime}(1)=-\pi / 2}, your limit is
 

\mathrm{-\frac{1}{-\pi / 2} \cdot 2 \cdot 1=\frac{4}{\pi}}
If using the derivative is not allowed, then use a substitution: for
\mathrm{\lim _{x \rightarrow 1} \frac{1-x}{\cos (\pi x / 2)}}
\mathrm{set \: 1-x=2 t / \pi\: so \: x=1-2 t / \pi\: and}
\mathrm{\frac{\pi}{2} x=\frac{\pi}{2}-t}

so the limit becomes
\mathrm{\lim _{t \rightarrow 0} \frac{2 t / \pi}{\cos (\pi / 2-t)}=\lim _{t \rightarrow 0} \frac{2}{\pi} \frac{\sin t}{t}}

Posted by

Sanket Gandhi

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