An asymptote is a straight line that approaches a curve indefinitely but does not intersect at any point. There are asymptotes at t [tex]\dfrac{3}{2}\rm\ and\ \dfrac{-1}{3}[/tex].
An asymptote is a straight line that approaches a curve indefinitely but does not intersect at any point. In other words, when a curve approaches infinity, it approaches an asymptote line.
The function that is given to us is:
[tex]f(x)=\dfrac{x+1}{6x^2-7x-3}[/tex]
As in the function, there is a fraction therefore, the denominator should not be zero in order for the function to be defined. Therefore, there are asymptotes at
[tex]6x^2-7x-3 = 0\\[/tex]
Substituting the value in the formula of the quadratic equation we will get,
[tex]D=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\D = \dfrac{-(-7)\pm\sqrt{(-7)^2-4(6)(-3)}}{2(6)}\\\\D =\dfrac{-(-7)\pm\sqrt{11}}{12} = \dfrac{3}{2}, \dfrac{-1}{3}[/tex]
Thus, there are asymptotes at [tex]\dfrac{3}{2}\rm\ and\ \dfrac{-1}{3}[/tex].
Learn more about Asymptotes:
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