Evaluating $\lim_{x\to 0} \frac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{x^2}$

Question

I want to evaluate:

$$\lim_{x\to 0} \frac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{x^2}.$$

Here's what I did. We know that as soon as ${x\to 0}$ $$1 - \cos x = \frac{x^2}{2} + O(x^4)$$ Therefore $$\cos x = 1 - \frac{x^2}{2} + O(x^4)$$ Now we apply this to cosines in limit $$\cos 2x = 1 - 2x^2 + O(x^4)$$ $$\cos 3x = 1 - \frac{9x^2}{2} + O(x^4)$$ Then use equality $$ (1 + x)^n = 1 + xn + o(x)$$ It yields us $$\lim_{x\to 0} \frac{1 - (1 - \frac{x^2}{2})(1 - x^2)(1 - \frac{3x^2}{2})}{x^2}$$ After simplification 1s cancel and we get $-(-1/2 - 1 - 3/2) = 3$.

Answer 1

Since in a neighbourhood of the origin $\cos(x)= 1-\frac{x^2}{2}+o(x^2)$, it follows that $\cos(2x)= 1-2x^2+o(x^2)$ and $\cos(3x)= 1-\frac{9x^2}{2}+o(x^2)$. Since $\sqrt{1+z}=1+\frac{z}{2}+o(z)$ and $\sqrt[3]{1+z}=1+\frac{z}{3}+o(z)$ we get that: $$ \cos(x)= 1-\frac{x^2}{2}+o(x^2),\quad \sqrt{\cos(2x)}= 1-x^2+o(x^2),\quad \sqrt[3]{\cos(3x)}= 1-\frac{3x^2}{2}+o(x^2) $$ hence their product behaves like $1-\left(\frac{1}{2}+1+\frac{3}{2}\right)x^2=1-3x^2$ and the wanted limit is $\color{red}{\large 3}$.

The neglected parts in the above manipulations are $o(x^2)$, hence they do not affect the final outcome.

Answer 2

More generally, for any integer $N\geq 1$, $$\lim_{x\to 0}\frac{1-\prod_{n=1}^N \sqrt[n]{\cos(nx)}}{x^2}=\frac{N(N+1)}{4},$$ because $$\prod_{n=1}^N \sqrt[n]{\cos(nx)}=\prod_{n=1}^N \sqrt[n]{1+\frac{(nx)^2}{2}+o(x^2)}=\prod_{n=1}^N \left(1+\frac{(nx)^2}{n2}+o(x^2)\right)\\ =\prod_{n=1}^N \left(1+\frac{nx^2}{2}+o(x^2)\right)= 1+\frac{x^2}{2}\sum_{n=1}^N n+o(x^2)=1+\frac{N(N+1)}{4}\cdot x^2+o(x^2).$$

Answer 3

When we do the Tayor series expansion of the numerator, we don't care about the $x^3$ and higher powered terms. They are all going to 0. I am going to drop them as I go.

$\lim_\limits{x\to 0}\frac {1 - (1 - \frac {x^2}{2})(1-2x^2)^\frac12(1-\frac {9x^2}{2})^\frac 13}{x^2}$

Now do the binomial expansion and continue to drop the $x^3$ and higher powered terms.

$\lim_\limits{x\to 0}\frac {1 - (1 - \frac {x^2}{2})(1-x^2)(1-\frac {3x^2}{2})}{x^2}$

Multiply it out, (still dropping high powered terms)

$\lim_\limits{x\to 0}\frac {1 - (1 - (\frac 12+1+\frac 32) x^2)}{x^2}$

And evaluate.

$ 3$

Answer 4

The $x^2$ at the denominator suggests we can use the fact that $$ \lim_{x\to0}\frac{1-\cos x}{x^2}=\frac{1}{2} $$ Rewrite the numerator as $$ (1-\sqrt{\cos2x}\sqrt[3]{\cos3x})+\sqrt{\cos2x}\sqrt[3]{\cos3x}(1-\cos x) $$ Since $$ \lim_{x\to0}\frac{\sqrt{\cos2x}\sqrt[3]{\cos3x}(1-\cos x)}{x^2}= \frac{1}{2} $$ we are reduced to compute $$ \lim_{x\to0}\frac{1-\sqrt{\cos2x}\sqrt[3]{\cos3x}}{x^2} $$ Let's retry the trick, rewriting the numerator as $$ (1-\sqrt[3]{\cos3x})+\sqrt[3]{\cos3x}(1-\sqrt{\cos2x}\,) $$ Now $$ \lim_{x\to0}\frac{1-\sqrt{\cos2x}}{x^2}= \lim_{x\to0}4\frac{1-\cos2x}{(2x)^2}\frac{1}{1+\sqrt{\cos2x}}= 4\cdot\frac{1}{2}\cdot\frac{1}{2}=1 $$ This leaves us with $$ \lim_{x\to0}\frac{1-\sqrt[3]{\cos3x}}{x^2}= \lim_{x\to0}9\frac{1-\cos3x}{(3x)^2}\frac{1}{1+\sqrt[3]{\cos3x}+\sqrt[3]{\cos^23x}}=9\cdot\frac{1}{2}\cdot\frac{1}{3}=\frac{3}{2} $$ Thus our limit is $$ \frac{3}{2}+1+\frac{1}{2}=3 $$

Alternative method. Define $$ g_n(t)=\log\sqrt[n]{\cos(n\sqrt{t})}=\frac{1}{n}\log\cos(n\sqrt{t}) $$ defined in a suitable neighborhood of $0$.

For $t\ne0$, the derivative is $$ g_n'(t)= \frac{1}{n} \frac{1}{\cos(n\sqrt{t})} (-\sin(n\sqrt{t}))\frac{n}{2\sqrt{t}} $$ and the limit at $0$ is easily seen to be $-n/2$, so by l'Hôpital the function $g_n$ is also differentiable at $0$.

The function you have to compute the limit of is even, so we can as well compute the limit from the right and do the substitution $x=\sqrt{t}$, so we have $$ \lim_{t\to0^+}\frac{1-\cos\sqrt{t}\sqrt{\cos(2\sqrt{t})}\sqrt[3]{\cos(3\sqrt{t})}}{t} $$ which is the negative of the derivative at $0$ of the function $$ f(t)=\cos\sqrt{t}\sqrt{\cos(2\sqrt{t})}\sqrt[3]{\cos(3\sqrt{t})} $$ and we can do the logarithmic derivative, so $$ \log f(t)=g_1(t)+g_2(t)+g_3(t) $$ and so $$ \frac{f'(t)}{f(t)}=g_1'(t)+g_2'(t)+g_3'(t) $$ Since $f(0)=1$, our limit is $$ -f'(0)=-(g_1'(t)+g_2'(t)+g_3'(t))=\frac{1}{2}+\frac{2}{2}+\frac{3}{2}=3 $$