Evaluate the Given limit: \(\lim_{x\rightarrow 0} x\,sec\,x\)

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 x · sec x

Step 1: Rewrite sec x

sec x = 1 / cos x

Step 2: Apply limits

As x → 0,
x → 0
cos x → 1

Step 3: Evaluate the limit

lim_x→0 x · sec x = 0 × 1

= 0

Final Answer:

lim_x→0 x · sec x = 0

Evaluate the Given limit: \(\lim_{x\rightarrow 0} x\,sec\,x\)

Solution and Explanation

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