Find $\lim\limits_{x\rightarrow \infty }\dfrac{5x^{4}-3x^{2}}{2x^{2}+1}$.

Figure 58 $\lim\limits_{x\rightarrow \infty } R(x)=\infty$

Solution $R( x) =\dfrac{5x^{4}-3x^{2}}{2x^{2}+1}$ is a rational function defined for all real numbers. Find the limit by dividing each term of the numerator and the denominator by the term with the highest power of $x$ that appears in the denominator, in this case, $2x^{2}$. Then $$\begin{eqnarray*} \lim\limits_{x\rightarrow \infty }\dfrac{5x^{4}-3x^{2}}{2x^{2}+1} \underset{\underset{\underset{\color{#0066A7}{\hbox{and denominator by \(2x^{2}\)}}}{\color{#0066A7}{\hbox{Divide the numerator}}}}{\color{#0066A7}{\uparrow}}}{=}\lim\limits_{x\rightarrow \infty }\dfrac{ \dfrac{5x^{4}-3x^{2}}{2x^{2}}}{\dfrac{2x^{2}+1}{2x^{2}}}=\lim\limits_{x \rightarrow \infty }\dfrac{\dfrac{5x^{2}}{2}-\dfrac{3}{2}}{1+\dfrac{1}{2x^{2} }}=\infty \end{eqnarray*}$$

because $\dfrac{5x^{2}}{2}-\dfrac{3}{2}$ $\rightarrow \infty$ and $1+\dfrac{ 1}{2x^{2}}$ $\rightarrow$ $1$ as $x\rightarrow \infty$.