Mathematics

Problem

Consider the function

g

defined for all real

x

g(x)=\sqrt{x^{2}+4x}-x

. (a) Rewrite

g(x)

in an algebraically equivalent form that is more convenient for analyzing its end behavior. Show all steps. (b) Determine

\displaystyle \lim_{x\to\infty} g(x)

. (c) Determine

\displaystyle \lim_{x\to -\infty} g(x)

. Carefully address the sign of each term involved. (d) State the equation(s) of all horizontal asymptote(s) of

y=g(x)

, providing complete justification based on your results from parts (b) and (c).