Question about the limit $\lim\limits_{+\infty} \tfrac{x^4}{1+x^4(\cos(x))^2}$ and result given by walpha

I have a problem evaluating this limit

$$\lim\limits_{x\to +\infty} \dfrac{x^4}{1+x^4(\cos(x))^2}$$

I'm still not able to get value Wolframalpha gives : $2$.

but if this was true then the reciprocal of this function should tend to $\dfrac 1 2$ but the reciprocal is

$$\dfrac{1}{x^4} + \cos^2(x)$$

which has no limit since first term tends to $0$ and second has no limit.

Where am I mistaken ? thanks for help.

Take the sequence $x_{n}=\frac{(2n+1)\pi}{2}$.

Then along this sequence the limit as $n\to\infty$ is $+\infty$.

But along the sequence $x_{n}=2n\pi$.

The limit is $1$.

These two different values are sufficient to show that the limit at infinity does not exist .