What is the oblique asymptote of #y = ( x^3 + 5x^2 + 3x + 10 )/( x^2 + 1 )#?

1 Answer
Sep 2, 2015

Answer:

The oblique asymptote is #color(red)(y = x +5)#

Explanation:

#y = (x^3+5x^2+3x+10)/(x^2+1)#

A slant (oblique) asymptote occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator.

To find the slant asymptote you must divide the numerator by the denominator.

I will use synthetic division:

#" "" "|1" "5" "3" "10#
#color(white)(1)-1|" "color(white)(1)-1color(white)(1)-5#
#0" "color(white)(1)|" "0" "0#
#" "" "stackrel("—————————)#
#" "" "color(white)(1)1" "5" "color(red)(2" "color(white)(1)5)#

The quotient is #x+5# with a remainder of #2x+5#.

We can ignore the remainder, so the oblique asymptote is #y=x+5#.

Graph