Certainly, let’s solve these limit problems step-by-step.

Problem 1:

$\lim_{{x \to 3}} \frac{x^2 - 7x + 12}{x^2 - 9x}$

Step-by-Step Solution:

1. Factor the Numerator and Denominator: $x^2 - 7x + 12 = (x - 3)(x - 4)$ $x^2 - 9x = x(x - 9)$

2. Rewrite the Expression: Substitute the factored forms: $\frac{(x - 3)(x - 4)}{x(x - 9)}$

3. Evaluate the Limit by Substitution: As $x \to 3$, substituting $x = 3$ directly would make the denominator zero, creating an indeterminate form. To address this, we simplify the expression further.

4. Simplify and Attempt to Cancel Out Terms: Unfortunately, no further simplification is possible here, so this limit doesn't cancel cleanly.