Answer
$$
\lim\limits_{x \to \infty} \frac{\sqrt {9x^{2}+5}}{2x}
$$
(a)
A graphing calculator can give a deceptive view of a function. The Figure shows the graph appears to have horizontal asymptotes at $y=\pm 1.5 $. Thus we determine that :
$$
\lim\limits_{x \to \infty} \frac{\sqrt {9x^{2}+5}}{2x}=1.5
$$
(b)
As $x \rightarrow \infty$
$$
\frac{\sqrt{9 x^{2}+5}}{2 x} \rightarrow \frac{3|x|}{2 x}
$$
since $x>0,|x|=x,$ so
$$
\frac{3|x|}{2 x}=\frac{3 x}{2 x}=\frac{3}{2}
$$
and, to evaluate the limit at infinity of a rational function, divide the numerator and denominator by the largest power of the variable that appears in the denominator, $x$ here, and then use these results. Thus, we find that
\[
\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{2}+5}}{2 x}=\frac{3}{2} \text { or } 1.5
\]
Work Step by Step
$$
\lim\limits_{x \to \infty} \frac{\sqrt {9x^{2}+5}}{2x}
$$
(a)
A graphing calculator can give a deceptive view of a function. The Figure shows the graph appears to have horizontal asymptotes at $y=\pm 1.5 $. Thus we determine that :
$$
\lim\limits_{x \to \infty} \frac{\sqrt {9x^{2}+5}}{2x}=1.5
$$
(b)
As $x \rightarrow \infty$
$$
\frac{\sqrt{9 x^{2}+5}}{2 x} \rightarrow \frac{3|x|}{2 x}
$$
since $x>0,|x|=x,$ so
$$
\frac{3|x|}{2 x}=\frac{3 x}{2 x}=\frac{3}{2}
$$
and, to evaluate the limit at infinity of a rational function, divide the numerator and denominator by the largest power of the variable that appears in the denominator, $x$ here, and then use these results. Thus, we find that
\[
\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{2}+5}}{2 x}=\frac{3}{2} \text { or } 1.5
\]
