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Find the vertical asymptotes and removable discontinuities of the graph of $\text{\hspace{0.17em}}f(x)=\frac{{x}^{2}-25}{{x}^{3}-6{x}^{2}+5x}.$
Removable discontinuity at $\text{\hspace{0.17em}}x=5.\text{\hspace{0.17em}}$ Vertical asymptotes: $\text{\hspace{0.17em}}x=0,\text{}x=1.$
While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.
There are three distinct outcomes when checking for horizontal asymptotes:
Case 1: If the degree of the denominator>degree of the numerator, there is a horizontal asymptote at $\text{\hspace{0.17em}}y=0.$
In this case, the end behavior is $\text{\hspace{0.17em}}f(x)\approx \frac{4x}{{x}^{2}}=\frac{4}{x}.\text{\hspace{0.17em}}$ This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function $\text{\hspace{0.17em}}g(x)=\frac{4}{x},\text{\hspace{0.17em}}$ and the outputs will approach zero, resulting in a horizontal asymptote at $\text{\hspace{0.17em}}y=0.\text{\hspace{0.17em}}$ See [link] . Note that this graph crosses the horizontal asymptote.
Case 2: If the degree of the denominator<degree of the numerator by one, we get a slant asymptote.
In this case, the end behavior is $\text{\hspace{0.17em}}f(x)\approx \frac{3{x}^{2}}{x}=3x.\text{\hspace{0.17em}}$ This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function $\text{\hspace{0.17em}}g(x)=3x.\text{\hspace{0.17em}}$ As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of $\text{\hspace{0.17em}}g(x)=3x\text{\hspace{0.17em}}$ looks like a diagonal line, and since $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ will behave similarly to $\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ it will approach a line close to $\text{\hspace{0.17em}}y=3x.\text{\hspace{0.17em}}$ This line is a slant asymptote.
To find the equation of the slant asymptote, divide $\text{\hspace{0.17em}}\frac{3{x}^{2}-2x+1}{x-1}.\text{\hspace{0.17em}}$ The quotient is $\text{\hspace{0.17em}}3x+1,\text{\hspace{0.17em}}$ and the remainder is 2. The slant asymptote is the graph of the line $\text{\hspace{0.17em}}g(x)=3x+1.\text{\hspace{0.17em}}$ See [link] .
Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at $\text{\hspace{0.17em}}y=\frac{{a}_{n}}{{b}_{n}},\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}{a}_{n}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{b}_{n}\text{\hspace{0.17em}}$ are the leading coefficients of $\text{\hspace{0.17em}}p\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}q\left(x\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}f(x)=\frac{p(x)}{q(x)},q(x)\ne 0.$
In this case, the end behavior is $\text{\hspace{0.17em}}f(x)\approx \frac{3{x}^{2}}{{x}^{2}}=3.\text{\hspace{0.17em}}$ This tells us that as the inputs grow large, this function will behave like the function $\text{\hspace{0.17em}}g(x)=3,\text{\hspace{0.17em}}$ which is a horizontal line. As $\text{\hspace{0.17em}}x\to \pm \infty ,f(x)\to 3,\text{\hspace{0.17em}}$ resulting in a horizontal asymptote at $\text{\hspace{0.17em}}y=3.\text{\hspace{0.17em}}$ See [link] . Note that this graph crosses the horizontal asymptote.
Notice that, while the graph of a rational function will never cross a vertical asymptote , the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.
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