How does the rate constant change with the change of the temperature? In class, we discussed the following multiple-choice question.

The rate expression for a reaction is:
$$rate = k[X][Y]$$
Which statement is correct?
A. As the temperature increases, the rate constant decreases.
B. The rate constant increases with increased temperature, but eventually reaches a constant value.
C. As the temperature increases, the rate constant increases.
D. The rate constant is not affected by a change in temperature.

Considering the Arrhenius equation given below, the rate constant is dependent on the absolute temperature of the reactants and activation energy of the reaction.
$$k = Ae^{-\frac{E_\mathrm{A}}{RT}}$$
Taking the following limit as the absolute temperature approaches infinity reveals that the rate constant converges to the Arrhenius constant at very high temperatures. This is also evident when graphing the Arrhenius equation. Therefore, we conclude that option B is correct.
$$\lim_{T \to \infty} Ae^{-\frac{E_\mathrm{A}}{RT}} = A\lim_{T \to \infty} e^{-\frac{E_\mathrm{A}}{RT}} = A$$
However, when checking the mark-scheme, option C is listed as the correct answer. Is there a mistake in my reasoning?
Answer B is technically correct although C could also be accepted. Assuming $E_A$ to be constant, at low temperatures because of the inverse $T$ in the exponential $\exp(-E_A/(R\cdot \text{small number}) \equiv \exp(-\text{big})$ which is a small number. At high temperatures $\exp(-E_A/(R\cdot \text{big number})\equiv \exp(-\text{small})$ is a big number, so the rate constant increases with temperature. 

At temperature such that the exponential  $\to 1$ the rate constant is $A$ for all practical purposes. Usually this limit is not reached unless $E_A$ is very small and then as $A$ is also a function of temperature this ($A$) term becomes now important. Generally, however, for the vast majority of reactions the exponential term is the most important.