Answer
$$\frac{1}{2}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = x - \sqrt {x\left( {x - 1} \right)} \cr
& {\text{Evaluate }}f\left( x \right){\text{ for the given values and complete the table}}{\text{.}} \cr
& x = {10^0} \to f\left( {{{10}^0}} \right) = {10^0} - \sqrt {{{10}^0}\left( {{{10}^0} - 1} \right)} = 1 \cr
& x = {10^1} \to f\left( {{{10}^1}} \right) = {10^1} - \sqrt {{{10}^1}\left( {{{10}^1} - 1} \right)} \approx 0.51316 \cr
& x = {10^2} \to f\left( {{{10}^2}} \right) = {10^2} - \sqrt {{{10}^2}\left( {{{10}^2} - 1} \right)} \approx 0.50125 \cr
& x = {10^3} \to f\left( {{{10}^3}} \right) = {10^3} - \sqrt {{{10}^3}\left( {{{10}^3} - 1} \right)} \approx 0.50012 \cr
& x = {10^4} \to f\left( {{{10}^4}} \right) = {10^4} - \sqrt {{{10}^4}\left( {{{10}^4} - 1} \right)} \approx 0.50001 \cr
& x = {10^5} \to f\left( {{{10}^5}} \right) = {10^5} - \sqrt {{{10}^5}\left( {{{10}^5} - 1} \right)} \approx 0.50000 \cr
& x = {10^6} \to f\left( {{{10}^6}} \right) = {10^6} - \sqrt {{{10}^6}\left( {{{10}^6} - 1} \right)} \approx 0.50000 \cr} $$
\[\boxed{\begin{array}{*{20}{c}}
x&{f\left( x \right)} \\
{{{10}^0}}&1 \\
{{{10}^1}}&{0.51316} \\
{{{10}^2}}&{0.50125} \\
{{{10}^3}}&{0.50012} \\
{{{10}^4}}&{0.50001} \\
{{{10}^5}}&{0.50000} \\
{{{10}^6}}&{0.50000}
\end{array}}\]
$$\eqalign{
& \mathop {\lim }\limits_{x \to \infty } \left[ {x - \sqrt {x\left( {x - 1} \right)} } \right] \cr
& {\text{Rationalizing}} \cr
& {\text{ = }}\mathop {\lim }\limits_{x \to \infty } \left[ {\frac{{x - \sqrt {x\left( {x - 1} \right)} }}{1} \times \frac{{x + \sqrt {x\left( {x - 1} \right)} }}{{x + \sqrt {x\left( {x - 1} \right)} }}} \right] \cr
& {\text{ = }}\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{{x^2} - {{\left( {\sqrt {x\left( {x - 1} \right)} } \right)}^2}}}{{x + \sqrt {x\left( {x - 1} \right)} }}} \right) \cr
& {\text{ = }}\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{{x^2} - x\left( {x - 1} \right)}}{{x + \sqrt {{x^2} - x} }}} \right) \cr
& {\text{ = }}\mathop {\lim }\limits_{x \to \infty } \left( {\frac{x}{{x + \sqrt {{x^2} - x} }}} \right) \cr
& {\text{ = }}\mathop {\lim }\limits_{x \to \infty } \left( {\frac{1}{{1 + \sqrt {\frac{{{x^2}}}{{{x^2}}} + \frac{x}{{{x^2}}}} }}} \right) \cr
& {\text{ = }}\mathop {\lim }\limits_{x \to \infty } \left( {\frac{1}{{1 + \sqrt {1 + \frac{1}{x}} }}} \right) \cr
& {\text{Evaluate the limit}} \cr
& {\text{ = }}\frac{1}{{1 + \sqrt {1 + \frac{1}{\infty }} }} \cr
& {\text{ = }}\frac{1}{{1 + \sqrt 1 }} \cr
& = \frac{1}{2} \cr
& \cr
& {\text{Graph}} \cr} $$
