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  • The value of  <img alt="\begin{array}{r} \lim _{x \rightarrow 1} 512 \pi(1-x) \\ \\\tan \left(\frac{\pi x}{2}\right) \end{array}" src="https://entrancecorner.oncodecogs.com/gif.latex

The value of

 \begin{array}{r} \lim _{x \rightarrow 1} 512 \pi(1-x) \\ \\\tan \left(\frac{\pi x}{2}\right) \end{array}   must be 

Option: 1

1010


Option: 2

1020


Option: 3

1024


Option: 4

850


Answers (1)

best_answer

\because \lim _{x \rightarrow 1}(1-x) \tan \left(\frac{\pi x}{2}\right) \\\\\text{Put}x=1+h \\\\\text{then} \lim _{h \rightarrow 0}(-h) \tan \left(\frac{\pi}{2}(1+h)\right)

                             \begin{aligned} & =-\lim _{h \rightarrow 0} h\left(-\cot \frac{\pi h}{2}\right) \\ \\& =\lim _{h \rightarrow 0} \frac{h}{\tan \left(\frac{\pi h}{2}\right)} \end{aligned}

                             =\frac{2}{\pi} \cdot \lim _{h \rightarrow 0} \frac{\frac{\pi h}{2}}{\tan \left(\frac{\pi h}{2}\right)}=\frac{2}{\pi} \times 1=\frac{2}{\pi}

\therefore \lim _{x \rightarrow 1} 512 \pi(1-x) \tan \left(\frac{\pi x}{2}\right)=512 \pi \times \frac{2}{\pi}=1024

Posted by

Nehul

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