MCV4U1 –UNIT FOUR Lesson Three
Lesson Three: Vertical, Horizontal, and Slant (Oblique) Asymptotes
Vertical Asymptotes:
The graph of a function f(x) has a vertical asymptote at x=a if lim f  x    . Vertical asymptotes occur when
xa
the denominator equals zero.
Ex. f  x  
1
3
x 4
2
Horizontal Asymptotes:
The graph of a function f(x) has a horizontal asymptote at y=L if lim f  x   L . Horizontal asymptotes are
x  
found by taking the limit of the function as x approaches infinity.
4 x 2  3x  1
Ex. f x  
2x 2  8
MCV4U1 –UNIT FOUR Lesson Three
Slant, or oblique, asymptotes:
In general, y=mx+b is a slant, or oblique, asymptote if the vertical distance between the curve y=f(x) and the
line approaches zero as x gets large positive or negative.
A graph can have a slant asymptote when
lim f x  mx  b 0
x
For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree
of the denominator. We can find the slant asymptote by division.
Ex. Find the slant asymptote of the function f x  
x 2

x 2  1 x 3  2x 2
x  2 


 lim  f x   mx  b lim  x  2  2
 x  2
x
x 
x  1


1 2 
 2
00
 x  2 
 lim 2
 lim  x x  
0
1  1 0
x x  1
x 
 1 2 

x 

x3  x
x 3  2x 2
x2 1
2x 2  x
2x 2  2
x 2
 y  x  2 is the slant asymptote
 f x   x  2 
x 2
x2 1

MCV4U1 –UNIT FOUR Lesson Three