If $\lim _{x \rightarrow 0} \frac{\cos (2 x)+a \cos (4 x)-b}{x^{4}}$ is finite, then $(a+b)$ is equal to :

JEE Main · 2025 · 02 Apr 2025 · Shift 2 · Limits

lim_{x \to 0} \frac{c o s ( 2 x ) + a c o s ( 4 x ) - b}{x ^{4}}

is finite, then

(a + b)

is equal to :

Answer: (A)

Expanding,

cos 2 x = 1 - 2 x^{2} + \frac{2}{3} x^{4} - \dots

and

cos 4 x = 1 - 8 x^{2} + \frac{32}{3} x^{4} - \dots

, so the numerator is

(1 + a - b) + (- 2 - 8 a) x^{2} + (\frac{2}{3} + \frac{32 a}{3}) x^{4} + \dots

. For the limit over

x^{4}

to be finite the constant and the

x^{2}

coefficient must vanish:

- 2 - 8 a = 0

gives

a = - \frac{1}{4}

, and

1 + a - b = 0

gives

b = \frac{3}{4}

. Hence

a + b = \frac{1}{2}

JEE Main · 2025 · 02 Apr 2025 · Shift 2 · Single correct option

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