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} ) (a) calculate ( f(x) ) for each value of ( x ) in the following table.

Explanation:

Step1: Simplify the function

We know that \(x^{2}-36=(x - 6)(x + 6)\). So \(f(x)=\frac{x^{2}-36}{x - 6}=\frac{(x - 6)(x + 6)}{x - 6}=x + 6\) for \(x
eq6\).

Step2: Calculate \(f(x)\) for \(x = 5.9\)

Substitute \(x = 5.9\) into \(f(x)=x + 6\), we get \(f(5.9)=5.9+6=11.9\).

Step3: Calculate \(f(x)\) for \(x = 5.99\)

Substitute \(x = 5.99\) into \(f(x)=x + 6\), we get \(f(5.99)=5.99+6=11.99\).

Step4: Calculate \(f(x)\) for \(x = 5.999\)

Substitute \(x = 5.999\) into \(f(x)=x + 6\), we get \(f(5.999)=5.999+6=11.999\).

Step5: Calculate \(f(x)\) for \(x = 5.9999\)

Substitute \(x = 5.9999\) into \(f(x)=x + 6\), we get \(f(5.9999)=5.9999+6=11.9999\).

Step6: Calculate \(f(x)\) for \(x = 6.1\)

Substitute \(x = 6.1\) into \(f(x)=x + 6\), we get \(f(6.1)=6.1+6=12.1\).

Step7: Calculate \(f(x)\) for \(x = 6.01\)

Substitute \(x = 6.01\) into \(f(x)=x + 6\), we get \(f(6.01)=6.01+6=12.01\).

Step8: Calculate \(f(x)\) for \(x = 6.001\)

Substitute \(x = 6.001\) into \(f(x)=x + 6\), we get \(f(6.001)=6.001+6=12.001\).

Step9: Calculate \(f(x)\) for \(x = 6.0001\)

Substitute \(x = 6.0001\) into \(f(x)=x + 6\), we get \(f(6.0001)=6.0001+6=12.0001\).

Answer:

\(x\)	\(5.9\)	\(5.99\)	\(5.999\)	\(5.9999\)	\(6.1\)	\(6.01\)	\(6.001\)	\(6.0001\)

For part (b), from the table, as \(x\) approaches \(6\) (both from the left - hand side \(x<6\) and the right - hand side \(x > 6\)), \(f(x)\) approaches \(12\). So \(\lim_{x
ightarrow6}\frac{x^{2}-36}{x - 6}=12\).