Finding the Limit of a Rational Function

Find:

1. \(\lim\limits_{x\rightarrow 1}\dfrac{3x^{3}-2x+1}{4x^{2}+5}\)
2. \(\lim\limits_{x\rightarrow -2}\dfrac{2x+4}{3x^{2}-1}\)

Solution

1. Since \(1\) is in the domain of the rational function \(R( x) =\dfrac{3x^{3}-2x+1}{4x^{2}+5},\) \[ \begin{equation*} \lim_{x\rightarrow 1}R( x) \underset{\underset{\color{#0066A7}{\hbox{Use (3)}}}{\color{#0066A7}{\uparrow}}}{=}R( 1) =\frac{3-2+1}{4+5}=\frac{2}{9} \end{equation*} \]
2. Since \(-2\) is in the domain of the rational function \(H(x) =\dfrac{2x+4}{3x^{2}-1}\), \[ \begin{equation*} \lim_{x\rightarrow -2}H( x) \underset{\underset{\color{#0066A7}{\hbox{Use (3)}}}{\color{#0066A7}{\uparrow}}}{=}H( -2) =\frac{-4+4}{12-1}=\dfrac{0}{11}=0 \end{equation*} \]