Cosine Everywhere

Calculus Level 3

$\displaystyle \lim_{x \to 0} \frac { \sqrt{\cos x} - \sqrt[3]{\cos x} }{\sin^2 x} .$

Evaluate the closed form of the limit above to 3 significant figures.

Details and assumptions

The answer to this Calculus problem is a real number.

The answer is -0.08333.

1 solution

Jatin Yadav
Dec 14, 2013

$L=\displaystyle \lim_{x \to 0} \frac{\sqrt{1 -\frac{x^2}{2!} + \frac{x^4}{4!} \dots} - \sqrt[3] {1 - \frac{x^2}{2!} + \frac{x^4}{4!} \dots}}{x^2}$ ( Using $\displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1$ )

Using the approximation $(1+x)^n \approx 1 + nx$ , if $x <<1$ , and ignoring unimp. terms,

We get $L = \lim_{x \to 0} \frac{ 1 - \frac{x^2}{4} - 1 + \frac{x^2}{6}}{x^2} = \boxed{\frac{-1}{12}}$

$- \frac {1}{12}$ is a special number. Coincidence? I think not.

Sharky Kesa - 7 years, 1 month ago

Really nice solution! I just used the L'Hôpital's rule

Carlos David Nexans - 6 years, 10 months ago

Cosine Everywhere

The answer is -0.08333.

1 solution

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