Question

find (limlimits_{x \to 4} \frac{x^2 + 3x - 28}{x - 4}). options: 11, dne, 0, 3

Explanation:

Step1: Factor the numerator

We factor the quadratic expression in the numerator \(x^{2}+3x - 28\). We need two numbers that multiply to \(- 28\) and add up to \(3\). The numbers are \(7\) and \(-4\), so \(x^{2}+3x - 28=(x + 7)(x-4)\). So the limit becomes \(\lim_{x
ightarrow4}\frac{(x + 7)(x - 4)}{x - 4}\).

Step2: Cancel the common factor

Since \(x
ightarrow4\) but \(x
eq4\) (we are taking the limit, not evaluating at \(x = 4\)), we can cancel the common factor \((x - 4)\) from the numerator and the denominator. This gives us \(\lim_{x
ightarrow4}(x + 7)\).

Step3: Evaluate the limit

Now we substitute \(x = 4\) into the expression \(x+7\). So \(4 + 7=11\).

Answer:

11