f(x)={[sin(x)," for "0

Question

f(x)={[sin(x)," for "0 <= x <= pi],[(x)/( pi)-1," for "pi < x <= 10]:} Find  lim_(x rarr pi)f(x). Choose 1 answer: (A) -1 (B) 0 (C)  pi (D) The limit doesn't exist.

Bytelearn · Accepted Answer

Separating the limits: To find the limit of the piecewise function f(x) as x approaches pi, we need to consider the limit from the left and the limit from the right separately.Limit from the left: First, let's find the limit from the left, which means we are approaching pi from values less than pi. For this part of the function, f(x) = sin(x). The limit as x approaches pi from the left of sin(x) is sin(pi).Limit from the right: We know that sin(pi) = 0. Therefore, the limit of f(x) as x approaches pi from the left is 0.Comparison of limits: Now, let's find the limit from the right, which means we are approaching pi from values greater than pi. For this part of the function, f(x) = (x/pi) - 1. The limit as x approaches pi from the right of (x/pi) - 1 is (pi/pi) - 1.Overall limit: We know that (pi/pi) - 1 simplifies to 1 - 1, which equals 0. Therefore, the limit of f(x) as x approaches pi from the right is also 0.Final answer: Since the limit from the left and the limit from the right both exist and are equal, the overall limit of f(x) as x approaches pi exists and is equal to the common value of the one-sided limits.The limit of f(x) as x approaches pi is 0. Therefore, the correct answer is (B) 0.

\[ f(x)=\left\{\begin{array}{ll} \sin (x) & \text { for } 0 \leq x \leq \pi \\ \frac{x}{\pi}-1 & \text { for } \pi

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