Find the vertical , horizontal and oblique asymptotes of `q(x)= (5x^2-13x-6)/( 2x^2-5x-3)`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The asymptotes of q(x) = (5x^2 - 13x - 6)/(2x^2 - 5x - 3) have to be determined.

The vertical asymptotes are given by x = a where a is the zeros of the denominator.

2x^2 - 5x - 3 = 0

=> 2x^2 - 6x + x - 3 = 0

=> 2x(x - 3) + 1(x - 3) = 0

=> (2x + 1)(x - 3) = 0

=> x = -1/2 and x = 3

The horizontal asymptote is given by y = (leading coefficient of numerator)/(leading coefficient of denominator). This gives y = 5/2

As the numerator and denominator have the same degree there is no slant asymptote.