IIT-JEE 2009 · Paper 1 · Mathematics Q30

← IIT-JEE 2009 · Paper 1

MathematicsMultiple correct+4 / -1Question 30 of 60

Let
$L=\lim_{x\to0}\frac{a-\sqrt{a^2-x^2}-\dfrac{x^2}4}{x^4},\qquad a>0.$
If $L$ is finite, then

a=2

a=1

L=\dfrac1{64}

L=\dfrac1{32}

Answer and worked solution

Official answer: A, C

Expand. $\sqrt{a^2-x^2}=a-\dfrac{x^2}{2a}-\dfrac{x^4}{8a^3}-\cdots$ , so the numerator is
$\left(\frac1{2a}-\frac14\right)x^2+\frac{x^4}{8a^3}+\cdots.$

Finite limit needs the $x^2$ term to vanish: $a=2$ . Then $L=\dfrac1{8\cdot8}=\dfrac1{64}$ .

(A), (C)

Topics:Limits Binomial expansion

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IIT-JEE 2009 · Paper 1, Mathematics question 30