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Answer :
The limit as x approaches 0 from the left for the function f(x) is 7, the limit as x approaches 0 from the right is 0, and the overall limit at x = 0 does not exist since the limits from both sides are not equal.
The student is asking for the limit of a piecewise function as x approaches 0 from the left (negative side) and from the right (positive side), as well as the overall limit at x = 0. We have two different functions for f(x) based on whether x is less than 0 or greater than or equal to 0.
For x < 0, we use f(x) = (5x - 6x² + 7). To find the limit as x approaches 0 from the negative side, we simply plug in 0 for x since there's no division by x that could cause issues. Hence, lim_(x
ightarrow0^(-)) f(x) = 7.
For x \">= 0, we use f(x) = -7sin(x). Since the sine function is continuous, and sin(0) = 0, the limit as x approaches 0 from the positive side is lim_(x
ightarrow0^(+)) f(x) = -7sin(0) = 0.
To determine if lim_(x
ightarrow0) f(x) exists, we need to check if the limits from both sides are equal. Since the left-hand limit is 7 and the right-hand limit is 0, these are not equal, and therefore the overall limit does not exist.
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