It seems like you're asking for the limit of a function as \( x \) approaches \( 0.2 \). Let's break down the expression you provided:

\[
f(x) = \frac{e^{2.8} - e^{1.3x}}{x}
\]
and you want to evaluate the limit as \( x \to 0.2 \).

The limit expression would be:

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

### Step-by-step solution:

1. **Substitute** \( x = 0.2 \) into the expression:

   \[
   f(0.2) = \frac{e^{2.8} - e^{1.3 \times 0.2}}{0.2}
   \]

2. Calculate the exponents:
   \[
   e^{2.8} \quad \text{and} \quad e^{1.3 \times 0.2} = e^{0.26}
   \]

Let me compute the values and finish the calculation.After computing the values, we find:

\[
\lim_{{x \to 0.2}} \frac{e^{2.8} - e^{1.3x}}{x} \approx 75.74
\]

Would you like a deeper explanation or have any questions?

Here are 5 related questions:
1. How do limits behave when approaching different values of \( x \)?
2. What is the significance of exponential functions in limits?
3. Can you explain the concept of continuity using limits?
4. How does the substitution method simplify limit evaluation?
5. How can we apply L'Hopital's rule to indeterminate forms?

**Tip:** Always check if the direct substitution works when evaluating limits before considering advanced techniques like L'Hopital's rule.

Explore how to find the limit of the function f(x) = (e^2.8 - e^1.3x)/x as x approaches 0.2. This problem involves key mathematical concepts such as limits, exponential functions, and the substitution method. Detailed step-by-step solution provided. Suitable for Grades 11-12.

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