find the horizontal and vertical asymptotes of the curve. you may want comma - separated lists. if an answer…

find the horizontal and vertical asymptotes of the curve. you may want comma - separated lists. if an answer does not exist, enter dne.)\n$y = \\frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 8 x - 3 }$

find the horizontal and vertical asymptotes of the curve. you may want comma - separated lists. if an answer does not exist, enter dne.)\n$y = \\frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 8 x - 3 }$

Answer

Answer:

Vertical asymptotes: (x = \frac{1}{3}), (x=-3); Horizontal asymptote: (y=\frac{2}{3})

Explanation:

Step1: Find vertical asymptotes

Set denominator (3x^{2}+8x - 3=0). Factor: ((3x - 1)(x + 3)=0). Solve (3x-1 = 0) gives (x=\frac{1}{3}), solve (x + 3=0) gives (x=-3).

Step2: Find horizontal asymptotes

Since degree of numerator (n = 2) and degree of denominator (m=2) ( (n=m) ). Use formula (y=\frac{a_{n}}{b_{m}}), where (a_{n}) is leading coefficient of numerator ((a_{n}=2)) and (b_{m}) is leading coefficient of denominator ((b_{m}=3)). So (y=\frac{2}{3}).