Evaluate the Given limit: \(\lim_{x\rightarrow 0}\) \(\frac{cos\,x}{\pi-x}\)

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:

lim_x→0 cos x / (π − x)

Step 1: Apply limits to numerator and denominator

As x → 0,

cos x → 1
π − x → π

Step 2: Evaluate the limit

lim_x→0 cos x / (π − x)

= 1 / π

Final Answer:

lim_x→0 cos x / (π − x) = 1 / π

Evaluate the Given limit: \(\lim_{x\rightarrow 0}\) \(\frac{cos\,x}{\pi-x}\)

Solution and Explanation

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