# Chapter 2 - Section 2.6 - Limits at Infinity; Horizontal Asymptotes - 2.6 Exercises - Page 139: 53

By graphing the function on the interval $-10 \leq x \leq 10$, we would guess that the horizontal asymptote is $y = 1$ $\lim\limits_{x \to \infty}f(x) = 3$ $\lim\limits_{x \to -\infty}f(x) = 3$ The horizontal asymptote is $y = 3$ The interval $-10 \leq x \leq 10$ does not give us the true indication of the horizontal asymptote because this function approaches the asymptote $y=3$ very gradually. $f(x) = \frac{3x^3+500x^2}{x^3+500x^2+100x+2000}$ By graphing the function on the interval $-10 \leq x \leq 10$, we would guess that the horizontal asymptote is $y = 1$ We can evaluate the limit as $x$ approaches infinity: $\lim\limits_{x \to \infty}\frac{3x^3+500x^2}{x^3+500x^2+100x+2000}$ $= \lim\limits_{x \to \infty}\frac{3x^3/x^3+500x^2/x^3}{x^3/x^3+500x^2/x^3+100x/x^3+2000/x^3}$ $= \frac{3}{1}$ $= 3$ Similarly, $\lim\limits_{x \to -\infty}f(x) = 3$ The horizontal asymptote is $y = 3$ The interval $-10 \leq x \leq 10$ does not give us the true indication of the horizontal asymptote because this function approaches the asymptote $y=3$ very gradually. 