Answer
\[\infty \]
Work Step by Step
\[\begin{gathered}
f\left( x \right) = \frac{{2{x^2}}}{{x + 1}} \hfill \\
{\text{Evaluate }}f\left( x \right){\text{ for the given values and complete the table}}{\text{.}} \hfill \\
x = {10^0} \to f\left( {{{10}^0}} \right) = \frac{{2{{\left( {{{10}^0}} \right)}^2}}}{{\left( {{{10}^0}} \right) + 1}} = 1 \hfill \\
x = {10^1} \to f\left( {{{10}^1}} \right) = \frac{{2{{\left( {{{10}^1}} \right)}^2}}}{{\left( {{{10}^1}} \right) + 1}} \approx 18.1818 \hfill \\
x = {10^2} \to f\left( {{{10}^2}} \right) = \frac{{2{{\left( {{{10}^2}} \right)}^2}}}{{\left( {{{10}^2}} \right) + 1}} \approx 198.02 \hfill \\
x = {10^3} \to f\left( {{{10}^3}} \right) = \frac{{2{{\left( {{{10}^3}} \right)}^2}}}{{\left( {{{10}^3}} \right) + 1}} \approx 1998.001 \hfill \\
x = {10^4} \to f\left( {{{10}^4}} \right) = \frac{{2{{\left( {{{10}^4}} \right)}^2}}}{{\left( {{{10}^4}} \right) + 1}} \approx 19998 \hfill \\
x = {10^5} \to f\left( {{{10}^5}} \right) = \frac{{2{{\left( {{{10}^5}} \right)}^2}}}{{\left( {{{10}^5}} \right) + 1}} \approx 199998 \hfill \\
x = {10^6} \to f\left( {{{10}^6}} \right) = \frac{{2{{\left( {{{10}^6}} \right)}^2}}}{{\left( {{{10}^6}} \right) + 1}} \approx 1999998 \hfill \\
\boxed{\begin{array}{*{20}{c}}
x&{f\left( x \right)} \\
{{{10}^0}}&1 \\
{{{10}^1}}&{18.1818} \\
{{{10}^2}}&{198.02} \\
{{{10}^3}}&{1998.001} \\
{{{10}^4}}&{19998} \\
{{{10}^5}}&{199998} \\
{{{10}^6}}&{1999998}
\end{array}} \hfill \\
{\text{Therefore,}} \hfill \\
\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \left( {\frac{{2{x^2}}}{{x + 1}}} \right) = \infty \hfill \\
{\text{Graph}} \hfill \\
\end{gathered} \]
