If is finite, then (a + b) is equal to : [2025]

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Solution

Q.
If $\lim_{x \to 0} \frac{\cos (2 x) + a \cos (4 x) - b}{x^{4}}$ is finite, then (a + b) is equal to : [2025]

1 $\frac{3}{4}$

2 –1

3 $\frac{1}{2}$

4 0

Ans.
(3)

$\lim_{x \to 0} \frac{\cos 2 x + a \cos 4 x - b}{x^{4}}$

$= \lim_{x \to 0} \frac{(1 - \frac{{(2 x)}^{2}}{2!} + \frac{{(2 x)}^{4}}{4!} . . .) + a (1 - \frac{{(4 x)}^{2}}{2!} + \frac{{(4 x)}^{4}}{4!} . . .) - b}{x^{4}}$

$= \lim_{x \to 0} \frac{(1 + a - b) + (- 2 - 8 a) x^{2} + (\frac{2}{3} + \frac{32}{3} a) x^{4} + . . . . . Higher power of x .}{x^{4}}$

Since limit is finite so, we have

1 + a – b = 0 and – 2 – 8a = 0

$\Rightarrow a = - \frac{1}{4} and b = 1 - \frac{1}{4} = \frac{3}{4}$

$∴ a + b = \frac{- 1}{4} + \frac{3}{4} = \frac{2}{4} = \frac{1}{2}$

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