Question

complete the parts below to determine the apparent value of the following limit.
lim _ { x
ightarrow 6 } \frac { 6 - 3 sqrt { 3 x - 14 } } { 6 - x }
(a) fill in the blanks. do not round intermediate computations, and round your answers to 4 decimal places where
applicable.

(b) use the values from part (a) to fill in the apparent value of the following limit.
lim _ { x
ightarrow 6 } \frac { 6 - 3 sqrt { 3 x - 14 } } { 6 - x } = square

Explanation:

Step1: Substitute \(x = 6.001\) into \(\frac{6 - 3\sqrt{3x-14}}{6 - x}\)

First, calculate \(3x-14\) when \(x = 6.001\): \(3\times6.001-14=18.003 - 14 = 4.003\). Then \(\sqrt{3x - 14}=\sqrt{4.003}\approx2.0007\). Next, \(6-3\sqrt{3x - 14}=6-3\times2.0007 = 6 - 6.0021=- 0.0021\). And \(6 - x=6 - 6.001=-0.001\). So \(\frac{6 - 3\sqrt{3x-14}}{6 - x}=\frac{-0.0021}{-0.001}=2.1\)

Step2: Substitute \(x = 6.1\) into \(\frac{6 - 3\sqrt{3x-14}}{6 - x}\)

Calculate \(3x-14\) when \(x = 6.1\): \(3\times6.1-14 = 18.3-14 = 4.3\). Then \(\sqrt{3x - 14}=\sqrt{4.3}\approx2.0736\). Next, \(6-3\sqrt{3x - 14}=6-3\times2.0736=6 - 6.2208=-0.2208\). And \(6 - x=6 - 6.1=-0.1\). So \(\frac{6 - 3\sqrt{3x-14}}{6 - x}=\frac{-0.2208}{-0.1}=2.208\)

Answer:

(a) For \(x = 6.001\), the value is \(2.1\); for \(x = 6.1\), the value is \(2.208\)
(b) \(\lim_{x
ightarrow6}\frac{6 - 3\sqrt{3x-14}}{6 - x}=2.25\)