Evaluate \(\lim_{x\rightarrow 5}\) f(x), where f(x) = |x|-5

Question

Let \( f(x) = \int \frac{(2 - x^2) \cdot e^x \cdot \sqrt{1 + x} \cdot (1 - x)^{3/2}}{dx} \). If \( f(0) = 0 \), then \( f\left(\frac{1}{2}\right) \) is equal to:

Answer 1

Given:

f(x) = |x| − 5

Step 1: Analyze the function near x = 5

Since 5 is a positive number,

|x| = x (for x near 5)

So, f(x) = x − 5

Step 2: Evaluate the limit

lim_x→5 f(x) = lim_x→5 (x − 5)

= 5 − 5

= 0

Final Answer:

lim_x→5 f(x) = 0

Evaluate \(\lim_{x\rightarrow 5}\) f(x), where f(x) = |x|-5

Solution and Explanation

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