Isa Trevino

2020-12-28

The vertical, horizontal and oblique asymptotes of the rational function $R\left(x\right)=\frac{6{x}^{2}+19x-7}{3x-1}$.

Clelioo

Given:
The rational function is defined by $R\left(x\right)=\frac{6{x}^{2}+19x-7}{3x-1}$.
Result used:
Vertical asymptote:
Consider a rational function $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$, in its lowest terms. If ris a real zero of the polynomial q(x), then x = r is a vertical asymptote of the function R.
Horizontal or oblique asymptotes:
Consider a rational function $R\left(x\right)=p\frac{x}{q}\left(x\right)=\frac{{a}_{n}{x}^{n}+{1}_{n-1}{x}^{n-1}+...{a}_{1}x+{a}_{0}}{{b}_{m}}{x}^{+}{b}_{m-1}{x}^{m-1}+...{b}_{1}x+{b}_{0}$ where nis the degree of the polynomial p(x) and mis the degree of the polynomial q(x). If $n=m+1$, the rational function R has no horizontal asymptote and only has an oblique asymptote given by $y=ax+b$ where $y=ax+b$ is quotient obtained by the polynomial division $\frac{p\left(x\right)}{q\left(x\right)}$.
Calculation:
Rewrite the rational function $R\left(x\right)=\frac{6{x}^{2}+19x-7}{3x-1}$ as follows.
$R\left(x\right)=\frac{6{x}^{2}+19x-7}{3x-1}$
$=\frac{6{x}^{2}+21x-2-7}{3x-1}$
$=3x\left(2x+7\right)-1\frac{2x+7}{3x-1}$
$=\frac{\left(2x+7\right)\left(3x-1\right)}{3x-1}$
$=2x+7$
Hence, the rational function $R\left(x\right)=\frac{6{x}^{2}+19x-7}{3x-1}$ in its lowest terms is $R\left(x\right)=2x+7$.
The polynomial in the denominator of $R\left(x\right)=2x+7$ is 1 and 1 has no zeros.
Therefore, the rational function $R\left(x\right)=\frac{6{x}^{2}+19x-7}{3x-1}$ has no vertical asymptote.
Note that, the degree of the polynomial in the numerator is 2 and the degree of the polynomial in the denominator is 1.
That is, the degree of the polynomial in the numerator is 1 more than the degree of the polynomial in the denominator.
Therefore, the rational function $R\left(x\right)=\frac{6{x}^{2}+19x-7}{3x-1}$ has no horizontal asymptote.
The polynomial division $\frac{6{x}^{2}+19x-7}{3x-1}$ gives the quotient as 2x +7.
Therefore, the rational function $\frac{6{x}^{2}+19x-7}{3x-1}$ has an oblique asymptote given by $y=2x+7$.
Hence, the rational function $\frac{6{x}^{2}+19x-7}{3x-1}$ has no vertical asymptote, no horizontal ext and oblique asymptote is $y=2x$.

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