Question

let ( f(x)=\frac{x^{2}-36}{x - 6} ). (a) calculate ( f(x) ) for each value of ( x ) in the following table. (b) make a conjecture about the value of ( lim _{x
ightarrow 6} \frac{x^{2}-36}{x - 6} )(type integers or decimals rounded to four decimal places as needed )(b) make a conjecture about the value of ( lim _{x
ightarrow 6} \frac{x^{2}-36}{x - 6} )( lim _{x
ightarrow 6} \frac{x^{2}-36}{x - 6}=square ) (type an integer or a decimal )

Explanation:

Step1: Simplify the function

We know that \(a^{2}-b^{2}=(a + b)(a - b)\). For \(f(x)=\frac{x^{2}-36}{x - 6}\), since \(x^{2}-36=(x + 6)(x - 6)\), then \(f(x)=\frac{(x + 6)(x - 6)}{x - 6}\). When \(x
eq6\), we can cancel out the \(x - 6\) terms, and \(f(x)=x + 6\).

Step2: Calculate the limit

We want to find \(\lim_{x
ightarrow6}\frac{x^{2}-36}{x - 6}\). Using the simplified form \(f(x)=x + 6\) (for \(x
eq6\)), we substitute \(x = 6\) into \(x+6\). So \(\lim_{x
ightarrow6}\frac{x^{2}-36}{x - 6}=\lim_{x
ightarrow6}(x + 6)\).
Substituting \(x = 6\) into \(x + 6\), we get \(6+6=12\).

Answer:

12