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EXAMPLE 11Finding the Limit of a Rational Function

Find:

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

Solution

  1. (a) 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. (b) 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*}