Find all values of x, where the graph of
y = [(5x^3)-(6x)+(8)] / (x^2)
crosses its oblique asymptote.
Let f(x)=(5*x^3-6*x+8)/(x^2)
The oblique asymptote can be found by evaluating the limit
Lim f(x) as x->±∞
which can be obtained by dividing the leading term of the numerator (5x³) by that of the denominator (x²).
Thus y=5x³/x³=5x is the oblique asymptote.
To find intersections, equate
f(x) = 5x
and solve for x in
(5*x^3-6*x+8)/(x^2)=5x
Cross multiply and cancel 5x³ to get
-6x+8=0
or
x=4/3 (one root).