Question

find the limit of the rational function a. as ( x
ightarrowinfty ) and b. as ( x
ightarrow-infty ). write ( infty ) or ( -infty ) where appro

( f(x)=\frac{x + 3}{x^{2}+18} )

a. ( lim_{x
ightarrowinfty}\frac{x + 3}{x^{2}+18}=square ) simplify your answer.)

Explanation:

Step1: Divide numerator and denominator by \(x^{2}\)

$$\lim_{x ightarrow\infty}\frac{x + 3}{x^{2}+18}=\lim_{x ightarrow\infty}\frac{\frac{x}{x^{2}}+\frac{3}{x^{2}}}{\frac{x^{2}}{x^{2}}+\frac{18}{x^{2}}}$$

Step2: Simplify the expression

$$=\lim_{x ightarrow\infty}\frac{\frac{1}{x}+\frac{3}{x^{2}}}{1 + \frac{18}{x^{2}}}$$

Step3: Apply the limit

As \(x
ightarrow\infty\), \(\lim_{x
ightarrow\infty}\frac{1}{x}=0\) and \(\lim_{x
ightarrow\infty}\frac{1}{x^{2}} = 0\).
$$=\frac{0 + 0}{1+0}$$

Answer:

\(0\)