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The problem requires finding the limit:

\[
\lim_{{x \to e}} \frac{{x^{20} - 3x}}{{x - e}}
\]

To solve this, we recognize it as a limit that can be approached using **L'Hôpital's Rule**, since we have a form \(\frac{0}{0}\) when directly substituting \(x = e\). L'Hôpital's Rule tells us to differentiate the numerator and the denominator separately.

### Step-by-Step Solution

1. **Differentiate the Numerator**: The numerator is \(f(x) = x^{20} - 3x\).
   - The derivative of \(x^{20}\) with respect to \(x\) is \(20x^{19}\).
   - The derivative of \(-3x\) is \(-3\).
   - Thus, \(f'(x) = 20x^{19} - 3\).

2. **Differentiate the Denominator**: The denominator is \(g(x) = x - e\).
   - The derivative of \(x\) is \(1\), and since \(e\) is a constant, its derivative is \(0\).
   - Thus, \(g'(x) = 1\).

3. **Apply L'Hôpital's Rule**: Substitute into the limit:
   \[
   \lim_{{x \to e}} \frac{{f(x)}}{{g(x)}} = \lim_{{x \to e}} \frac{{20x^{19} - 3}}{{1}} = 20e^{19} - 3
   \]

### Final Answer

The result of the limit is:
\[
\lim_{{x \to e}} \frac{{x^{20} - 3x}}{{x - e}} = 20e^{19} - 3
\]

### Conclusion
The correct answer is therefore:

**(C) \(20e^{19} - 3\)**.

Would you like further details on any of these steps, or do you have questions?

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Here are five related questions to expand your understanding:

1. How does L'Hôpital's Rule work for limits that involve \(\infty/\infty\) forms?
2. What would happen if we used Taylor series expansion around \(x = e\) to solve this limit?
3. How do we handle limits where higher-order terms dominate in the numerator?
4. When is it better to factorize expressions rather than using L'Hôpital’s Rule?
5. Could we apply a different method, like polynomial approximation, to find this limit?

**Tip:** Always check if a limit produces an indeterminate form before applying L'Hôpital's Rule.

Find the limit \( \lim_{{x \to e}} \frac{{x^{20} - 3x}}{{x - e}} \).

Learn how to find the limit \( \lim_{{x \to e}} \frac{{x^{20} - 3x}}{{x - e}} \) using L'Hôpital's Rule. This AP Calculus problem involves differentiating the numerator and denominator and applying limit rules. Ideal for high school students studying calculus.

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