The expression you provided is:

$\lim_{{x \to 13}} \frac{{4x - 13}}{{x - 13}}.$

Steps to solve:

1. Substitute $x = 13$ into the expression directly:

   $\frac{{4(13) - 13}}{{13 - 13}} = \frac{{52 - 13}}{{0}} = \frac{{39}}{{0}}.$

   The denominator becomes 0, indicating a potential undefined form or vertical asymptote. This suggests we need a more sophisticated approach, such as factoring or canceling terms.

2. Factor and Simplify: The numerator can be rewritten as:

   $4x - 13 = 4(x - 13) + 39.$

   Thus, the expression becomes:

   $\lim_{{x \to 13}} \frac{{4(x - 13)}}{{x - 13}} = \lim_{{x \to 13}} 4.$

   The $(x - 13)$ cancels out, leaving us with the constant value 4.

Final Answer:

$\lim_{{x \to 13}} \frac{{4x - 13}}{{x - 13}} = 4.$

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions to expand on this concept:

1. How can we handle limits that result in indeterminate forms like $\frac{0}{0}$?
2. What are the steps to apply L'Hopital's Rule for indeterminate limits?
3. Can limits always be solved by direct substitution?
4. What are one-sided limits, and how do they differ from two-sided limits?
5. How does factoring help simplify rational functions in limits?

Tip: Always check for factorable expressions when solving limits that give indeterminate forms.