complete the parts below to determine the apparent value of the following limit.\n lim _ { x \rightarrow 6 }…

complete the parts below to determine the apparent value of the following limit.\n lim _ { x \rightarrow 6 } \frac { 6 - 3 sqrt { 3 x - 14 } } { 6 - x } \n(a) fill in the blanks. do not round intermediate computations, and round your answers to 4 decimal places where\napplicable.\n\n(b) use the values from part (a) to fill in the apparent value of the following limit.\n lim _ { x \rightarrow 6 } \frac { 6 - 3 sqrt { 3 x - 14 } } { 6 - x } = square

complete the parts below to determine the apparent value of the following limit.\n lim _ { x \rightarrow 6 } \frac { 6 - 3 sqrt { 3 x - 14 } } { 6 - x } \n(a) fill in the blanks. do not round intermediate computations, and round your answers to 4 decimal places where\napplicable.\n\n(b) use the values from part (a) to fill in the apparent value of the following limit.\n lim _ { x \rightarrow 6 } \frac { 6 - 3 sqrt { 3 x - 14 } } { 6 - x } = square

Answer

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\rightarrow6}\frac{6 - 3\sqrt{3x-14}}{6 - x}=2.25)