Cross your limits 2

Calculus Level 4

lim ⁡ x → 0 e − x 2 / 2 − cos ⁡ x x 3 sin ⁡ x \large \lim_{x\to0} \dfrac{e^{-x^2/2} - \cos x}{x^3 \sin x}

If the limit above is equal to A A , find 1 A \dfrac1A .

6 0 8 12

Akash Pratap Singh by Akash Pratap Singh , 21, India

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2 solutions

Otto Bretscher
Apr 30, 2016

Consider the Taylor polynomial in the numerator: lim ⁡ x → 0 ( x 4 8 − x 4 4 ! + O ( x 6 ) x 4 × x sin ⁡ x ) = 1 12 \lim_{x \to 0}\left(\frac{\frac{x^4}{8}-\frac{x^4}{4!}+O(x^6)}{x^4}\times\frac{x}{\sin x}\right)=\frac{1}{12} so A = 12 A=\boxed{12}

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Apply L'hôpital rule 5 times and you have the solution. It may be an inefficient way, so I look forward to an elegant solution

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