Evaluating $\lim_{x\to0}{\frac{\sqrt[3]{\cos{4x}}-\sqrt[3]{\cos{5x}}}{1-\cos{3x}}}$ without L'Hopital's rule

Question

I'm currently struggling with this task:
$$\lim_{x\to0}{\frac{\sqrt[3]{\cos{4x}}-\sqrt[3]{\cos{5x}}}{1-\cos{3x}}}$$
By now I have only come up with the idea to use the formula of sum of cubes, so, the numerator would become $\cos{4x}-\cos{5x}$ and the denominator would be $3(1-\cos{3x})$:
$$\frac{\sqrt[3]{\cos{4x}}-\sqrt[3]{\cos{5x}}}{1-\cos{3x}} = \frac{(\sqrt[3]{\cos{4x}}-\sqrt[3]{\cos{5x}})*(\sqrt[3]{cos^2{4x}} + \sqrt[3]{\cos{4x}}*\sqrt[3]{\cos{4x}} + \sqrt[3]{\cos^2{5x}})}{(1-\cos{3x})*(\sqrt[3]{cos^2{4x}} + \sqrt[3]{\cos{4x}}*\sqrt[3]{\cos{4x}} + \sqrt[3]{\cos^2{5x}})} = \frac{\cos{4x}-\cos{5x}}{3(1-\cos{3x})}$$ In this expression it looks like $\cos{4x}-\cos{5x}$ and $(1-\cos{3x})$ increase with the same speed (if I can say it this way) and the limit is likely to be $\frac{1}{3}$.
Still I have no idea how to prove that without using L'Hopital's rule which is prohibited by the task.

Answer 1

Once you have rendered

$f(x)=\dfrac{\cos 4x -\cos 5x}{3(1-\cos 3x)}$

you then use the trigonometric sum-product relations to get

$\cos 4x -\cos 5x=2\sin(x/2)\sin(9x/2)$

$1-\cos 3x=\cos 0x - \cos 3x=2\sin^2(3x/2)$

In the latter case, of course, the sum-product relation reduces to a double-angle identity.

Thereby

$f(x)=\dfrac{\sin(x/2)\sin(9x/2)}{3\sin^2(3x/2)}$

Now apply the fact, which probably you have available, that $g(u)=(\sin u)/u\to 1$ as $u\to 0$. Thus

$\sin(x/2)=(x/2)g(x/2)$

and similarly for the other sine arguments. After substitution and cancellation of common factors in the numerator and denominator we end with

$f(x)=\dfrac{g(x/2)g(9x/2)}{3g^2(3x/2)}$

and when we set the $g$ factors to their limit of $1$ we get the overall limit $1/3$.

Answer 2

Hint

Use http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html

$$\dfrac{\cos2ax-\cos2bx}{1-\cos2cx}=\dfrac{2\sin(b-a)x\cdot\sin(b+a)x}{2\sin^2cx}$$

Answer 3

You can use the Taylor Expansion of $\cos(x)$.

The numerator is $\dfrac{(5x)^2-(4x)^2}{2!} + O\left({x^4}\right)$

The denominator is $\dfrac{(3x)^2}{3\cdot2!} + O\left({x^4}\right)$

Dividing by $x^2$ and using $3^2+4^2=5^2$ gives the limit as $1/3$.

Answer 4

Adding and subtracting $1$ in the numerator we can rewrite the expression under limit as $$\frac{1-\sqrt [3]{\cos 5x}}{1-\cos 3x}-\frac{1-\sqrt[3]{\cos 4x}} {1-\cos 3x}$$ and the first fraction above can be further rewritten as $$\frac{1-\sqrt[3]{\cos 5x}}{1-\cos 5x}\cdot\frac{1-\cos 5x}{(5x)^2}\cdot\frac{5^2}{3^2}\cdot\frac{(3x)^2}{1-\cos 3x}$$ so that it tends to $(1/3)(1/2)(25/9)(2)=25/27$. Similarly second fraction tends to $16/27$ and hence the desired limit is $9/27=1/3$.

The following standard limits are used in the above approach $$\lim_{x\to a} \frac{x^n-a^n} {x-a} =na^{n-1},\,\lim_{x\to 0}\frac{1-\cos x} {x^2}=\frac{1}{2}$$