The limit problem in the image is as follows:

$\lim_{x \to -13} \frac{x^2 - 169}{x + 13}$

Step 1: Factor the numerator

We notice that $x^2 - 169$ is a difference of squares. We can factor it as:

$x^2 - 169 = (x - 13)(x + 13)$

So the expression becomes:

$\lim_{x \to -13} \frac{(x - 13)(x + 13)}{x + 13}$

Step 2: Simplify the expression

We can cancel out the common factor of $x + 13$ from both the numerator and denominator (provided $x \neq -13$, which is valid for the limit):

$\lim_{x \to -13} (x - 13)$

Step 3: Evaluate the limit

Now, substitute $x = -13$ into the simplified expression:

$x - 13 = -13 - 13 = -26$

Therefore, the value of the limit is:

$\boxed{-26}$

Let me know if you need further details or explanations. Here are some related questions:

1. How do you identify when an expression can be factored?
2. What happens if a common factor doesn’t exist after factoring?
3. Can limits exist even if the function is undefined at the point of interest?
4. How would you handle a more complex polynomial in the numerator?
5. What if the denominator were also a quadratic expression?

Tip: Always check for factorable expressions, like differences of squares, as they can simplify limits involving indeterminate forms.