Question

use the table of values to estimate lim f(x), lim f(x), and lim f(x).
x→6⁻ x→6⁺ x→6

x	5.9	5.99	5.999	5.9999	6.0001	6.001	6.01	6.1
f(x)	8.9	8.99	8.999	8.9999	9.0001	9.001	9.01	9.1

lim f(x)=
x→6⁻

Explanation:

Step1: Analyze left - hand limit values

As \(x\) approaches \(6\) from the left (\(x\to6^{-}\)), the \(x\) - values are \(5.9,5.99,5.999,5.9999\). The corresponding \(f(x)\) - values are \(8.9,8.99,8.999,8.9999\).

Step2: Determine left - hand limit

As \(x\) gets closer and closer to \(6\) from the left, \(f(x)\) gets closer and closer to \(9\). So, \(\lim_{x\to6^{-}}f(x)=9\).

Step3: Analyze right - hand limit values

As \(x\) approaches \(6\) from the right (\(x\to6^{+}\)), the \(x\) - values are \(6.0001,6.001,6.01,6.1\). The corresponding \(f(x)\) - values are \(9.0001,9.001,9.01,9.1\).

Step4: Determine right - hand limit

As \(x\) gets closer and closer to \(6\) from the right, \(f(x)\) gets closer and closer to \(9\). So, \(\lim_{x\to6^{+}}f(x)=9\).

Step5: Determine overall limit

Since \(\lim_{x\to6^{-}}f(x)=\lim_{x\to6^{+}}f(x) = 9\), then \(\lim_{x\to6}f(x)=9\).

Answer:

\(\lim_{x\to6^{-}}f(x)=9\), \(\lim_{x\to6^{+}}f(x)=9\), \(\lim_{x\to6}f(x)=9\)