It seems like you're asking to evaluate the limit of the function:

$\lim_{{x \to 0.2}} \frac{e^{2.8x} - e^{1.3x}}{x}$

Let's break this down:

We need to find the value of the limit at $x = 0.2$.

The function is $\frac{e^{2.8x} - e^{1.3x}}{x}$, and we want to substitute $x = 0.2$ into this expression.

Substitution:

1. Substitute $x = 0.2$ into the exponentials:

   \quad \text{and} \quad e^{1.3 \times 0.2} = e^{0.26}$$

2. Calculate the exponentials $e^{0.56}$ and $e^{0.26}$, then compute the difference between these two values.

3. Finally, divide the result by $x = 0.2$.

Let me calculate this.The value of the limit is approximately:

$\lim_{{x \to 0.2}} \frac{e^{2.8x} - e^{1.3x}}{x} \approx 2.27$

Would you like any further clarification or details on this solution?

Here are five related questions to expand on this topic:

1. How do you evaluate limits of more complex exponential functions?
2. What is the significance of using L'Hopital's rule when evaluating limits?
3. How do limits behave when exponential functions are involved?
4. Can you explain how the derivative of an exponential function is calculated?
5. How do limits relate to the concept of continuity?

Tip: Always check if direct substitution works first when evaluating limits. If indeterminate forms appear, consider using L'Hopital's Rule or simplifying the expression.