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

Isa Trevino 2020-12-28 Answered
The vertical, horizontal and oblique asymptotes of the rational function R(x)=6x2+19x73x1.
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Expert Answer

Clelioo
Answered 2020-12-29 Author has 88 answers
Given:
The rational function is defined by R(x)=6x2+19x73x1.
Result used:
Vertical asymptote:
Consider a rational function R(x)=p(x)q(x), 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(x)=pxq(x)=anxn+1n1xn1+...a1x+a0bmx+bm1xm1+...b1x+b0 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 p(x)q(x).
Calculation:
Rewrite the rational function R(x)=6x2+19x73x1 as follows.
R(x)=6x2+19x73x1
=6x2+21x273x1
=3x(2x+7)12x+73x1
=(2x+7)(3x1)3x1
=2x+7
Hence, the rational function R(x)=6x2+19x73x1 in its lowest terms is R(x)=2x+7.
The polynomial in the denominator of R(x)=2x+7 is 1 and 1 has no zeros.
Therefore, the rational function R(x)=6x2+19x73x1 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(x)=6x2+19x73x1 has no horizontal asymptote.
The polynomial division 6x2+19x73x1 gives the quotient as 2x +7.
Therefore, the rational function 6x2+19x73x1 has an oblique asymptote given by y=2x+7.
Hence, the rational function 6x2+19x73x1 has no vertical asymptote, no horizontal ext and oblique asymptote is y=2x.
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