It will simply be infinite.I have attached a graph that shows the behavior of the function near - infinity. And it is diverging!I have also simplified the function and clearly in diverges as x approaches infinity.

lim x→ -∞ [ √ ( x^2 - 2x) + x ] = lim x→ -∞ [ √ ( x^2 - 2x) + x ]* [ √ ( x^2 - 2x) - x ] / [ √ ( x^2 - 2x) - x ] = lim x→ -∞ [ ( x^2 - 2x) - x^2 ] / [ √ ( x^2 - 2x) - x ] = lim x→ -∞ - 2x / (- x)*[ √ ( 1 - 2/x) + 1 ] ..........(Taking √x^2 = (-x) in this case)= lim x→ -∞ 2 / [ √ ( 1 - 2/x) + 1 ] = lim x→ -∞ 2 / [ √ ( 1 - 2/x) + 1 ] = 2 / [ √ ( 1 - 0) + 1 ] = 1So lim x→ -∞ [ √ ( x^2 - 2x) + x ] = 1