IGNOU BMTC 102 2026 - Find the values of a and b, if

Question

Find the values of a and b, if

$$\lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x^3} = 1$

09 Jan 2026

Answer :

Word Count : 268

We are asked to find (a) and (b) such that [ \lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x^3} = 1. ] First, expand (\cos x) and (\sin x) using their Maclaurin series up to the (x^3) term: [ \cos x = 1 - \frac{x^2}{2} + \dots, \quad \sin x = x - \frac{x^3}{6} + \dots ] Substitute these into the numerator: [ x(1 + a \cos x) - b \sin x = x(1 + a(1 - \frac{x^2}{2})) - b(x - \frac{x^3}{6}) ] [ = x(1 + a - \frac{a x^2}{2}) -

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