To evaluate the limit:

$\lim_{x \to 4} e^{2x} + 2$

Step-by-step solution:

1. Substitute $x = 4$ directly into the expression, as this is a continuous function: $e^{2x} + 2$

2. Evaluate $e^{2x}$ at $x = 4$: $e^{2 \cdot 4} = e^8$

3. Add 2 to the result: $e^8 + 2$

Thus, the limit is: $\boxed{e^8 + 2}$

Would you like a deeper explanation, such as how continuity justifies the substitution?

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Expanding Questions:

1. What are the key properties of the exponential function $e^x$?
2. How does continuity simplify solving limits involving exponential functions?
3. Can $e^{2x}$ ever approach a finite value as $x \to \infty$?
4. What would happen if the function included $\ln(x)$ instead of $e^{2x}$?
5. Can this technique be used for piecewise functions?

Tip:

For exponential functions like $e^{kx}$, limits at finite points are often computed by direct substitution because the exponential function is continuous everywhere.