Question

$\lim\limits_{x \to 12} \frac{x^2 - 20x + 96}{x - 12} = $ (no calculator)

Explanation:

Step1: Factor the numerator

We factor the quadratic expression \(x^2 - 20x + 96\). We need two numbers that multiply to \(96\) and add up to \(-20\). Those numbers are \(-12\) and \(-8\), so \(x^2 - 20x + 96=(x - 12)(x - 8)\).
The limit becomes \(\lim_{x
ightarrow12}\frac{(x - 12)(x - 8)}{x - 12}\).

Step2: Cancel the common factor

Since \(x
ightarrow12\) but \(x
eq12\) (we are taking the limit, not evaluating at \(x = 12\)), we can cancel the common factor \((x - 12)\) from the numerator and the denominator.
After canceling, we get \(\lim_{x
ightarrow12}(x - 8)\).

Step3: Evaluate the limit

Now we substitute \(x = 12\) into the expression \(x - 8\). So \(12-8 = 4\).

Answer:

\(4\)