Arrhenius equation and rate constant
This is how the graph between the rate constant and temperature looks like:

This might not seem intuitive looking at the Arrhenius equation which says:
$$k=Ae^{\frac{-E_a}{RT}}$$
After using some mathematical prowess someone might say that as $T\rightarrow \infty, ~k\rightarrow A$ and so the graph should look something more like:

Which on first glance seems fair but doesn't match the aforementioned curve.
To understand what is causing the problem we must check everything that might change during a change in temperature.
Upon examination we can find that only two things might cause this discrepancy, $E_a$ and the pre-exponential &quot;constant&quot;, $A$.
The activation energy of a system is also independent of the temperature and thus the only possible problem point left is the pre-exponential &quot;constant&quot;. Turns out that this constant isn't a constant in true sense, which isn't that disturbing of an idea if you know what that constant signifies.
The pre-exponential constant shows how many effective collisions can take place in a reacting mixture. It is defined as,
$$A=pZ$$
Where,
$A$ is the pre-exponential constant.
$p$ is the steric factor, which takes into account the steric effects during an effective collision.
$Z$ is the collision frequency.
Now as it might seem intuitive, the collision frequency is related to temperature and increases with increase in temperature according to the following relation(using equations for collisions between similar molecules to prevent unnecessary complexity):
$$Z=4nr\sqrt{\frac{\pi RT}{M}}$$
Where,
$Z$ is collision frequency.
$n$ is molecules per unit volume.
$r$ is the radius of molecule.
$R$ is the universal gas constant.
$T$ is temperature.
$M$ is the molar mass of the gas.
Thus the collision frequency and corresponding the pre-exponential constant tend to infinity when $T\rightarrow \infty$, which matches with our observation of $k\rightarrow \infty$ when $T\rightarrow \infty$.
Reference:
The Effect of Temperature on Reaction Rates https://chem.libretexts.org/@go/page/24271 (accessed May 14, 2021).