Question

(lim_{x \to 17} \frac{x^2 - 3x - 238}{x - 17} = ) --- (no calculator)

Explanation:

Step1: Factor the numerator

We need to factor \(x^2 - 3x - 238\). We look for two numbers that multiply to \(-238\) and add up to \(-3\). The numbers are \(-17\) and \(14\) because \(-17\times14 = -238\) and \(-17 + 14=-3\). So, \(x^2 - 3x - 238=(x - 17)(x + 14)\).
The limit becomes \(\lim_{x
ightarrow17}\frac{(x - 17)(x + 14)}{x - 17}\)

Step2: Cancel the common factor

Since \(x
ightarrow17\) but \(x
eq17\) (we are taking the limit, not evaluating at \(x = 17\)), we can cancel the \((x - 17)\) terms in the numerator and the denominator.
We get \(\lim_{x
ightarrow17}(x + 14)\)

Step3: Evaluate the limit

Now we substitute \(x = 17\) into \(x + 14\). So, \(17+14 = 31\)

Answer:

\(31\)