We are asked to find the limit of the function $\frac{e^x - e^{-x}}{2e^x}$ as $x$ approaches $0$. That is, we need to calculate $C = \lim_{x \to 0} \frac{e^x - e^{-x}}{2e^x}$.

1. Problem Description

We are asked to find the limit of the function $\frac{e^x - e^{-x}}{2e^x}$ as $x$ approaches $0$.

That is, we need to calculate $C = \lim_{x \to 0} \frac{e^x - e^{-x}}{2e^x}$.

2. Solution Steps

We can evaluate the limit by substituting $x=0$ directly into the expression if it does not lead to an indeterminate form.

Substituting $x=0$ into the expression gives:

$\frac{e^0 - e^{-0}}{2e^0} = \frac{1-1}{2(1)} = \frac{0}{2} = 0$

Therefore, the limit is $0$.

3. Final Answer

$C = 0$