Given the following piecewise function, evaluate limx→−4+f(x).f(x)=−2×2−x−32x^2+x+32×2−2if x<−4 if −4≤x≤−2 if x>−2
To evaluate the limit limx→−4+f(x) of the given piecewise function, we need to determine the value of f(x) as x approaches -4 from both the left and the right
To evaluate the limit limx→−4+f(x) of the given piecewise function, we need to determine the value of f(x) as x approaches -4 from both the left and the right.
First, let’s consider the left-hand limit as x approaches -4:
limx→−4−f(x)
Since the function is defined as f(x) = -2x^2 – x – 3 for x < -4, we substitute x = -4 into this equation:
f(-4) = -2(-4)^2 - (-4) - 3
= -2(16) + 4 - 3
= -32 + 4 - 3
= -31
Next, we'll evaluate the right-hand limit as x approaches -4:
limx→−4+f(x)
As stated in the piecewise function, for -4 ≤ x ≤ -2, f(x) = -2. Therefore:
f(-4) = -2
Since the left-hand and right-hand limits evaluate to different values at x = -4, the limit does not exist.
This means that the function f(x) does not have a well-defined limit as x approaches -4. It behaves differently as x approaches -4 from the left and the right side.
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