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  • find the limit    <img alt="\mathrm{\lim _{x \rightarrow 2}\left(x^3-2 x-4\right) \tan \frac{\pi x}{4}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7

find the limit    \mathrm{\lim _{x \rightarrow 2}\left(x^3-2 x-4\right) \tan \frac{\pi x}{4}}

Option: 1

-\frac{40}{\pi}


Option: 2

0


Option: 3

1


Option: 4

None of these


Answers (1)

best_answer

Let  \mathrm{x=2+h} . As \mathrm{h \rightarrow 0} , using the asymptotic notation \mathrm{\sim}  and the fact that

lim \mathrm{_{t \rightarrow 0} \frac{\tan t-\tan 0}{t-0}=\tan ^{\prime}(0)=1}
\mathrm{x^3-2 x-4=10 h+6 h^2+h^3 \sim 10 h }


and


\mathrm{ \tan \frac{\pi x}{4}=-\frac{1}{\tan \frac{\pi h}{4}} \sim-\frac{4}{\pi h} . }
The limit of the product is therefore \mathrm{ -\frac{40}{\pi} }.

Posted by

SANGALDEEP SINGH

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