Mathematics

Problem

Define the piecewise function

f(x)=\begin{cases} \dfrac{2x^{2}+1}{x}, & x>1,\\ \dfrac{5x}{x^{2}+4}, & x\le 1. \end{cases}

Analyze the end behavior of

f

. (a) Evaluate

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

and justify each step of your computation. (b) Evaluate

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

, indicating clearly which branch of the definition is relevant and why. (c) State all horizontal asymptote equations for

f

, if any, and explain how each limit in parts (a) and (b) supports your conclusion. (d) Verify whether

f

is continuous at

x=1

. Explain how your result at

x=1

relates (or does not relate) to the existence of horizontal asymptotes.