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Hey can someone help me solve this? Thanks

$$\lim_{x \rightarrow 11 \pi/2} \frac{\cos (11 x)}{x - 11 \pi/2}$$

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Enclose it in dollar signs.. –  Your Ad Here Feb 4 at 0:13
    
I've edited your post to fix the formatting; please verify that it's correct. Can you please share what you've tried, and explain what you're having trouble with? –  T. Bongers Feb 4 at 0:15
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closed as off-topic by Austin Mohr, Ayman Hourieh, T. Bongers, Stefan Hansen, Michael Hoppe Feb 4 at 6:50

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2 Answers

Hint: Let $f(x)=\cos(11x)$. Note that $f(11\pi/2)=0$.

Thus we are looking for $$\lim_{x\to11\pi/2}\frac{f(x)-f(11\pi/2)}{x-11\pi/2}.$$ By definition, this is $f'(11\pi/2)$. Calculate the derivative using the ordinary rules of differentiation.

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Setting $\displaystyle x-\dfrac{11\pi}2=u$

$$\lim_{x\to\dfrac{11\pi}2}\left(\frac{\cos11x}{x-\dfrac{11\pi}2}\right)$$

$$=\lim_{u\to0}\left(\frac{\cos\left[11\left(u+\dfrac{11\pi}2\right)\right]}u\right)$$

$$=\lim_{u\to0}\left(\frac{\cos\left(11u+\dfrac{\pi}2\right)}u\right)$$

$$=-11\lim_{u\to0}\left(\frac{\sin11u}{11u}\right)$$ as $\cos\left(\frac\pi2+y\right)=-\sin y$

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