Let's analyze each statement in turn:

1. $\frac{x-4}{x^2 - 16} = \frac{1}{x + 4}$

This equation simplifies by factoring $x^2 - 16$ as $(x - 4)(x + 4)$, so: $\frac{x-4}{x^2 - 16} = \frac{x-4}{(x-4)(x+4)} = \frac{1}{x+4}$ As long as $x \neq 4$, this equation is valid. Therefore, the statement is true.

2. $\lim_{x \to 4} \left(\frac{x - 4}{x^2 - 16}\right) = \lim_{x \to 4} \left(\frac{1}{x + 4}\right) = \frac{1}{8}$

Using the simplification from statement 1, we know: $\lim_{x \to 4} \left( \frac{x - 4}{(x-4)(x+4)} \right) = \lim_{x \to 4} \left( \frac{1}{x + 4} \right) = \frac{1}{8}$ Thus, the statement is true.

3. $\lim_{x \to c} 5x^{10} = 5 \lim_{x \to c} x^{10}$

This statement refers to the property of limits where constants can be factored out of limits. The limit expression for powers is: $\lim_{x \to c} 5x^{10} = 5 \lim_{x \to c} x^{10}$ This property holds, so the statement is true.

4. $\lim_{x \to 1} \left( \frac{x^2 - 3x + 2}{x^2 + 4x - 5} \right)$ does not exist

Let's factor both the numerator and denominator: $x^2 - 3x + 2 = (x - 1)(x - 2), \quad x^2 + 4x - 5 = (x - 1)(x + 5)$ The expression simplifies to: $\frac{(x - 1)(x - 2)}{(x - 1)(x + 5)} = \frac{x - 2}{x + 5}$ As $x \to 1$, the limit becomes: $\frac{1 - 2}{1 + 5} = \frac{-1}{6}$ The limit exists and is equal to $-\frac{1}{6}$, so the statement is false.

5. $\lim_{x \to 4} \left[ f(x) g(x) \right] = f(4) g(4)$ if the limit exists

This is a standard property of limits. If the limits of $f(x)$ and $g(x)$ exist as $x \to 4$, then: $\lim_{x \to 4} \left[ f(x) g(x) \right] = f(4) g(4)$ The statement is true.

Conclusion:

The correct statements are:

1. $\frac{x-4}{x^2 - 16} = \frac{1}{x + 4}$
2. $\lim_{x \to 4} \left( \frac{x - 4}{x^2 - 16} \right) = \lim_{x \to 4} \left( \frac{1}{x + 4} \right) = \frac{1}{8}$
3. $\lim_{x \to c} 5x^{10} = 5 \lim_{x \to c} x^{10}$
4. $\lim_{x \to 4} \left[ f(x) g(x) \right] = f(4) g(4)$

The incorrect statement is: 4. $\lim_{x \to 1} \left( \frac{x^2 - 3x + 2}{x^2 + 4x - 5} \right)$ does not exist

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Would you like more details or have any further questions? Here are a few related questions to explore:

1. How do you handle limits involving indeterminate forms like $0/0$?
2. Can you explain how factoring helps in simplifying rational functions for limits?
3. What happens when you encounter limits involving infinity or large values of $x$?
4. What are some common properties of limits you use in solving problems?
5. How does L'Hopital's Rule apply to indeterminate forms like $0/0$?

Tip: Always check if a rational function can be simplified by factoring before taking limits!