Find the horizontal or oblique asymptote of f (x) = negative 3 x squared plus 7 x plus 1, all over x minus 2

Find the horizontal or oblique asymptote of f (x) = negative 3 x squared plus 7 x plus 1, all over x minus 2

Finding the horizontal and oblique asymptote of

f (x) = (-3x^2+7x+1) / (x-2)

Solution:

For Horizontal Asymptote:

Line y=L is a horizontal asymptote of the function y=f (x), if either lim x→∞ f (x) = L lim x→∞ f (x) = L or lim x→-∞ f (x) = L lim x→-∞ f (x) = L, and LL is finite.

Calulate limits:

lim x→∞ (1 x-2 (-3 x2 + 7x+1)) = - ∞

lim x→-∞ (1 x-2 (-3 x2 + 7x+1)) = ∞

Thus, there are no horizontal asymptotes.

For Oblique Asymptote:

Do polynomial long division (-3 x2 + 7x+1) / (x-2) = - 3x+1 + 3 / (x-2)

The rational term approaches 0 as the variable approaches infinity.

Thus, the slant asymptote is y=-3x+1y=-3x+1.