MHF 4U3 – ADVANCED FUNCTIONS
Unit 6 – Rational Functions
Lesson #26 – Slant Asymptotes
Slant Asymptotes (Oblique Asymptotes):
Slant asymptotes occur when the degree of the numerator is one more than the degree of the
denominator.
Slant asymptotes, like horizontal asymptotes, indicate the end behavior of a function.
Example 1: Find the equation of the slant asymptote for the function y =
Solution:
!! ! !!!!!
!!!
3x 2 − 7 x + 5
.
x+2
, 𝑥 ≠ −2
3𝑥 ! − 7𝑥 + 5
lim
= −∞
!→!!!
𝑥+2
3𝑥 ! − 7𝑥 + 5
=∞
!→!!
𝑥+2
⟹ 𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒂𝒔𝒚𝒎𝒑𝒕𝒐𝒕𝒆 𝒂𝒕 𝒙 = −𝟐
lim !
The numerator is one degree higher than the denominator ⇒ slant asymptote.
3 x − 13
x + 2 3x − 7 x + 5
2
3x 2 + 6 x
− 13 x + 5
− 13 x − 26
31
So y = 3x − 13 +
31
x+2
31
approaches zero and the values of f(x) approach 3x – 13.
x+2
This means the graph of f(x) is close to the graph of the line y = 3x – 13. This line is
31
called a slant (or oblique) asymptote. The remainder
tells us whether the
x+2
function is approaching the slant asymptote from above or below as x approaches
negative infinity and then positive infinity.
For large values of x,
!!
lim!→!! !!! = 0! ⇒ The function approaches the slant asymptote from below as
𝑥 → −∞
!!
lim!→! !!! = 0! ⇒ The function approaches the slant asymptote from above as 𝑥 → ∞
Homework: