Answer
$$\infty $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {x^2} - x\sqrt {x\left( {x - 1} \right)} \cr
& {\text{Evaluate }}f\left( x \right){\text{ for the given values and complete the table}}{\text{.}} \cr
& x = {10^0} \to f\left( {{{10}^0}} \right) = {\left( {{{10}^0}} \right)^2} - \sqrt {{{10}^0}\left( {{{10}^0} - 1} \right)} = 1 \cr
& x = {10^1} \to f\left( {{{10}^1}} \right) = {\left( {{{10}^1}} \right)^2} - \sqrt {{{10}^1}\left( {{{10}^1} - 1} \right)} \approx 5.1316 \cr
& x = {10^2} \to f\left( {{{10}^2}} \right) = {\left( {{{10}^2}} \right)^2} - \sqrt {{{10}^2}\left( {{{10}^2} - 1} \right)} \approx 50.1256 \cr
& x = {10^3} \to f\left( {{{10}^3}} \right) = {\left( {{{10}^3}} \right)^2} - \sqrt {{{10}^3}\left( {{{10}^3} - 1} \right)} \approx 500.1250 \cr
& x = {10^4} \to f\left( {{{10}^4}} \right) = {\left( {{{10}^4}} \right)^2} - \sqrt {{{10}^4}\left( {{{10}^4} - 1} \right)} \approx 5000.125 \cr
& x = {10^5} \to f\left( {{{10}^5}} \right) = {\left( {{{10}^5}} \right)^2} - \sqrt {{{10}^5}\left( {{{10}^5} - 1} \right)} \approx 50000.125 \cr
& x = {10^6} \to f\left( {{{10}^6}} \right) = {\left( {{{10}^6}} \right)^2} - \sqrt {{{10}^6}\left( {{{10}^6} - 1} \right)} \approx 500000.126 \cr} $$
\[\boxed{\begin{array}{*{20}{c}}
x&{f\left( x \right)} \\
{{{10}^0}}&1 \\
{{{10}^1}}&{5.1316} \\
{{{10}^2}}&{50.1256} \\
{{{10}^3}}&{500.1250} \\
{{{10}^4}}&{5000.125} \\
{{{10}^5}}&{50000.125} \\
{{{10}^6}}&{500000.126}
\end{array}}\]
$$\eqalign{
& \mathop {\lim }\limits_{x \to \infty } \left[ {{x^2} - x\sqrt {x\left( {x - 1} \right)} } \right] \cr
& {\text{Evaluate the limit}} \cr
& {\text{ = }}{\left( \infty \right)^2} - \infty \sqrt {\infty \left( {\infty - 1} \right)} \cr
& {\text{ = }}\infty \cr
& \cr
& {\text{Graph}} \cr} $$
