\lim_{x\to1 } \left ( 1+ \cos\pi x \right )\cot ^{2}\pi x

  • Option 1)

    -1

  • Option 2)

    \frac{1}{2}

  • Option 3)

    1

  • Option 4)

    None of these

 

Answers (1)

As we learnt in

Condition of Trigonometric Limits -

\lim_{x\rightarrow 0}\:\:\:\frac{sinx}{x}=1\:(it\:is\:less\:than\:1)


\lim_{x\rightarrow 0}\:\:\:\frac{tanx}{x}=1\:(it\:is\:more\:than\:1)

- wherein

because\:\:\:\:\:\frac{sinx}{x}<1

                    
                      \frac{tanx}{x}>1

 

 \lim_{x\rightarrow 1}(1+cos \pi x) cot^{2}\pi x

\lim_{x\rightarrow 1}2cos^{2}\frac{\pi x}{2}\times \frac{cos^{2}\pi x}{sin^{2}\pi x}

=\lim_{x\rightarrow 1} \frac{2cos^{2}\frac{\pi x}{2}\times cos^{2}\pi x}{4.sin^{2}\frac{\pi x}{2}.cos^{2}\frac{\pi x}{2}}

=\frac{1}{2}.\frac{1}{1}=\frac{1}{2}

 


Option 1)

-1

Incorrect

Option 2)

\frac{1}{2}

Correct

Option 3)

1

Incorrect

Option 4)

None of these

Incorrect

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