Answer
\[\frac{6}{{\sqrt 3 }}\]
Work Step by Step
\[\begin{gathered}
\mathop {\lim }\limits_{x \to \infty } \frac{{6x}}{{\sqrt {3{x^2} - 2x} }} \hfill \\
{\text{Evaluate }}f\left( x \right){\text{ for the given values and complete the table}}{\text{.}} \hfill \\
x = 1 \to f\left( 1 \right) = \frac{{6\left( 1 \right)}}{{\sqrt {3{{\left( 1 \right)}^2} - 2\left( 1 \right)} }} \approx 6 \hfill \\
x = 10 \to f\left( {10} \right) = \frac{{6\left( {10} \right)}}{{\sqrt {3{{\left( {10} \right)}^2} - 2\left( {10} \right)} }} \approx 3.5856 \hfill \\
x = {10^2} \to f\left( {{{10}^2}} \right) = \frac{{6\left( {{{10}^2}} \right)}}{{\sqrt {3{{\left( {{{10}^2}} \right)}^2} - 2\left( {{{10}^2}} \right)} }} \approx 3.4757 \hfill \\
x = {10^3} \to f\left( {{{10}^3}} \right) = \frac{{6\left( {{{10}^3}} \right)}}{{\sqrt {3{{\left( {{{10}^3}} \right)}^2} - 2\left( {{{10}^3}} \right)} }} \approx 3.4652 \hfill \\
x = {10^4} \to f\left( {{{10}^4}} \right) = \frac{{6\left( {{{10}^4}} \right)}}{{\sqrt {3{{\left( {{{10}^4}} \right)}^2} - 2\left( {{{10}^4}} \right)} }} \approx 3.4642 \hfill \\
x = {10^5} \to f\left( {{{10}^5}} \right) = \frac{{6\left( {{{10}^5}} \right)}}{{\sqrt {3{{\left( {{{10}^5}} \right)}^2} - 2\left( {{{10}^5}} \right)} }} \approx 3.4641 \hfill \\
\boxed{\begin{array}{*{20}{c}}
x&{f\left( x \right)} \\
1&6 \\
{10}&{3.5856} \\
{{{10}^2}}&{3.4757} \\
{{{10}^3}}&{3.4652} \\
{{{10}^4}}&{3.4642} \\
{{{10}^5}}&{3.4641}
\end{array}} \hfill \\
{\text{Therefore,}} \hfill \\
\mathop {\lim }\limits_{x \to \infty } \frac{{6x}}{{\sqrt {3{x^2} - 2x} }} \approx 3.4641 \hfill \\
{\text{Exact value :}} \hfill \\
\mathop {\lim }\limits_{x \to \infty } \frac{{6x}}{{\sqrt {3{x^2} - 2x} }} = \frac{6}{{\sqrt 3 }} \hfill \\
{\text{Graph}} \hfill \\
\end{gathered} \]
