Evaluate the Given limit: \(\lim_{x\rightarrow 0}(cosec\,x-cot\,x)\)

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 (cosec x − cot x)

Step 1: Use identity

cosec x − cot x = (1 − cos x) / sin x

Step 2: Rewrite the limit

lim_x→0 (1 − cos x) / sin x

Step 3: Multiply numerator and denominator by (1 + cos x)

= lim_x→0 [(1 − cos x)(1 + cos x)] / [sin x(1 + cos x)]

= lim_x→0 (1 − cos² x) / [sin x(1 + cos x)]

= lim_x→0 sin² x / [sin x(1 + cos x)]

= lim_x→0 sin x / (1 + cos x)

Step 4: Apply standard limits

As x → 0,
sin x → 0
cos x → 1

Therefore,

lim_x→0 sin x / (1 + cos x) = 0 / 2

= 0

Final Answer:

lim_x→0 (cosec x − cot x) = 0

Evaluate the Given limit: \(\lim_{x\rightarrow 0}(cosec\,x-cot\,x)\)

Solution and Explanation

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