Question

Evaluate the Given limit: \(\lim_{x\rightarrow 0}\) \(\frac{ax+xcos\,x}{b\,sin\,x}\)

Answer 1

Given:

lim_x→0 (ax + x cos x) / (b sin x)

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Step 1: Factor x from the numerator

= lim_x→0 x(a + cos x) / (b sin x)

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Step 2: Rewrite the expression

= (1/b) · lim_x→0 (a + cos x) · (x / sin x)

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Step 3: Apply standard limits

lim_x→0 (x / sin x) = 1

As x → 0,
cos x → 1

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Step 4: Evaluate the limit

lim_x→0 (a + cos x) · (x / sin x)

= (a + 1) · 1

= a + 1

Therefore,

lim_x→0 (ax + x cos x) / (b sin x)

= (a + 1) / b

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Final Answer:

lim_x→0 (ax + x cos x) / (b sin x) = (a + 1) / b