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NSW Y12 Maths - Advanced Graphs and Equations Vertical and Horizontal Asymptotes

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Vertical and Horizontal Asymptotes Theory

\textbf{Vertical asymptotes} are determined by letting the denominator in an algebraic fraction equal zero, and then solving the equations. The value of \(x\) is the required vertical asymptote.\\ \textbf{Horizontal asymptotes} are determined by dividing the numerator and denominator by the term with the highest power. Limits that go to zero are eliminated and the remaining number gives the horizontal asymptote.\\ \textbf{Example}\\ %11765 Consider the function \(f(x) = \dfrac{{2x + 1}}{{x - 2}}\) the vertical and horizontal asymptotes are?\\ \textbf{Solution}\\ $\begin{aligned} f(x) &=\frac{2 x+1}{x-2} \\ x-2 & \neq 0 \quad \rightarrow x=2\ \text{is a vertical asymptote.} \\ f(x) &=\frac{\frac{2 x}{x}+\frac{1}{x}}{\frac{x}{x}-\frac{2}{x}} \\ &=\frac{2+\frac{1}{x}}{1-\frac{2}{x}}\\ \text{As}\quad& x \rightarrow \infty \quad \frac{1}{x} \rightarrow 0, \frac{2}{x} \rightarrow 0 \\ \therefore\quad & f(x) \rightarrow 2 \\ \therefore\quad & f(x)=2 \text { is a horizontal asymptote } \end{aligned}$\\

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  • Vertical and Horizontal Asymptotes - Video - Finding Horizontal and Vertical Asymptotes Graphing Rational Functions

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