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:

$$
LATEXBLOCK0
$$

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:

\(21\)