Question

What is limit of (startstartfraction negative sine (startfraction pi x over 3 endfraction) overover 3 endendfraction) as x approaches 5? negative startfraction 3 startroot 3 endroot over 2 endfraction negative startfraction startroot 3 endroot over 6 endfraction startfraction startroot 3 endroot over 6 endfraction startfraction 3 startroot 3 endroot over 2 endfraction
answer: c

Answer 1

I'm guessing your limit is

$\displaystyle \lim_{x\to5} \frac{-\sin\left(\frac{\pi x}3\right)}3$

The limand is continuous at x = 5, so we can evaluate the limit directly by substituting x = 5:

$\displaystyle \lim_{x\to5} \frac{-\sin\left(\frac{\pi x}3\right)}3 = -\frac13 \sin\left(\frac{5\pi}3\right) = -\frac13 \times \left(-\frac{\sqrt3}2\right) = \boxed{\frac1{2\sqrt3}} = \boxed{\frac{\sqrt3}6}$