Question

lim\limits_{x\to 6} \frac{2x^2 - 3x - 54}{x - 6} = \\_\\_\\_ (no calculator)

Explanation:

Step1: Factor the numerator

We need to factor the quadratic expression \(2x^2 - 3x - 54\). We look for two numbers that multiply to \(2\times(-54)=-108\) and add up to \(-3\). The numbers are \(-12\) and \(9\). So we can rewrite the middle term:
\[

$$\begin{align*} 2x^2 - 3x - 54&=2x^2 - 12x + 9x - 54\\ &=2x(x - 6)+9(x - 6)\\ &=(2x + 9)(x - 6) \end{align*}$$

\]

Step2: Simplify the fraction

Now we substitute the factored form of the numerator into the limit expression:
\[
\lim_{x
ightarrow 6}\frac{(2x + 9)(x - 6)}{x - 6}
\]
Since \(x
ightarrow 6\) but \(x
eq 6\) (we are taking the limit, not evaluating at \(x = 6\)), we can cancel out the common factor \((x - 6)\) from the numerator and the denominator:
\[
\lim_{x
ightarrow 6}(2x + 9)
\]

Step3: Evaluate the limit

Now we can directly substitute \(x = 6\) into the expression \(2x+9\) because the function \(2x + 9\) is continuous everywhere (it's a polynomial):
\[
2(6)+9=12 + 9=21
\]

Answer:

\(21\)