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lim_(x rarr(pi)/(3))sin(x)=" ? " Choose 1 answer: (A) (1)/(2) (B) (sqrt3)/(2) (C) sqrt3 (D) The limit doesn't exist.

lim⁡x→π3sin⁡(x)= ?  \lim _{x \rightarrow \frac{\pi}{3}} \sin (x)=\text { ? } \newlineChoose 11 answer:\newline(A) 12 \frac{1}{2} \newline(B) 32 \frac{\sqrt{3}}{2} \newline(C) 3 \sqrt{3} \newline(D) The limit doesn't exist.

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Full solution

Q. lim⁡x→π3sin⁡(x)= ?  \lim _{x \rightarrow \frac{\pi}{3}} \sin (x)=\text { ? } \newlineChoose 11 answer:\newline(A) 12 \frac{1}{2} \newline(B) 32 \frac{\sqrt{3}}{2} \newline(C) 3 \sqrt{3} \newline(D) The limit doesn't exist.
  1. Identify the function: Identify the function whose limit needs to be found.\newlineWe need to find the limit of sin⁡(x)\sin(x) as xx approaches π3\frac{\pi}{3}.
  2. Determine continuity: Determine if the function is continuous at the point x=π3x = \frac{\pi}{3}. The sine function is continuous everywhere on the real number line, including at x=π3x = \frac{\pi}{3}.
  3. Evaluate at x=π3x = \frac{\pi}{3}: Evaluate the function at the point x=π3x = \frac{\pi}{3}. Since the sine function is continuous at x=π3x = \frac{\pi}{3}, we can find the limit by direct substitution. sin⁡(π3)=32\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
  4. Conclude the limit: Conclude the limit based on the calculation.\newlineThe limit of sin⁡(x)\sin(x) as xx approaches π3\frac{\pi}{3} is 32\frac{\sqrt{3}}{2}.

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