Rational functions
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For each of the following, give equations of the asymptotes of the graph \(y=f(x)\)
i) \(f(x)=\dfrac{x+5}{x-2}\)
ii) \(f(x)=\dfrac{x}{x^2+1}\)
\begin{align}
&\text{i)}\quad f(x)=\frac{x+5}{x-2}\\
&x-2 \neq 0 \rightarrow\ x \neq 2\\
&\therefore\ x=2 ~\text{is an asymptote }\\
&f(x)=\frac{x+5}{x-2}\\
&\text{Divide numerator and denominator by}\ x\\
&\begin{aligned}
f(x) &=\frac{\dfrac{x+5}{x}}{\dfrac{x-2}{x}} \\
&=\frac{1+\dfrac{5}{x}}{1-\dfrac{2}{x}}
\end{aligned}\\
&\text{as}\ x \rightarrow \infty \quad \frac{5}{x} \rightarrow 0,\quad -\frac{2}{x} \rightarrow 0 \\&\therefore f(x) \rightarrow 1 \\
&\therefore\ y=1\ \text{is an asymptote}\\ \\
&\text{ii)}\quad
f(x)=\frac{x}{x^{2}+1} \\
&x^{2}+1 \neq 0 \therefore \text { no asymptote } \\
&f(x)=\frac{\dfrac{x}{x^{2}}}{\dfrac{x^{2}}{x^{2}}+\dfrac{1}{x^{2}}}=\frac{\dfrac{1}{x}}{1+\dfrac{1}{x^{2}}} \\
&\text { As } x \rightarrow \infty \quad f(x) \rightarrow 0 \\
&\therefore y=0 \text { is an asymptote }
\end{align}
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