let ( f(x)=\frac{x^{2}-36}{x - 6} ). (a) calculate ( f(x) ) for each value of ( x ) in the following table…
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 \rightarrow 6} \frac{x^{2}-36}{x - 6} ) (a) calculate ( f(x) ) for each value of ( x ) in the following table.
Answer
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\neq6).
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) |
|---|---|---|---|---|---|---|---|---|
| (f(x)=\frac{x^{2}-36}{x - 6}) | (11.9) | (11.99) | (11.999) | (11.9999) | (12.1) | (12.01) | (12.001) | (12.0001) |
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\rightarrow6}\frac{x^{2}-36}{x - 6}=12).