JEE Main 2025 Apr 2 Shift 2, Mathematics Q14: Methods to Evaluate Limits

JEE Main2025Apr 2, Shift 2Mathematics

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

Solution

Expand each cosine as a Maclaurin series up to $x^{4}$ :

$cos (2 x) = 1 - 2 x^{2} + \frac{2}{3} x^{4} + O (x^{6})$ ,

$a cos (4 x) = a (1 - 8 x^{2} + \frac{32}{3} x^{4}) + O (x^{6})$ .

Numerator $= (1 + a - b) - (2 + 8 a) x^{2} + (\frac{2}{3} + \frac{32 a}{3}) x^{4} + O (x^{6})$ .

For the limit divided by $x^{4}$ to be finite, the $x^{0}$ and $x^{2}$ coefficients must vanish:

$1 + a - b = 0$ and $2 + 8 a = 0 \Rightarrow a = - \frac{1}{4}, b = 1 + a = \frac{3}{4}$ .

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

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