CSUN MATH 150A: Comprehensive Study Guide on Asymptotes
(Calculus I)
A CSUN Students Insight on Mastering Asymptotes
Welcome, fellow Math 150A student! Asymptotes are one of the most crucial
concepts youll encounter in Calculus I, as they connect limits (Chapter 2) with
curve sketching (Chapter 4). Dont just memorize the rules; understand why these
lines exist. An asymptote is a limiting behavior—its a line that a function approaches
but never truly touches as its input (x) or output (y) heads towards infinity.
Mastering this topic means mastering limits involving infinity, which is fundamental
to all future calculus courses. Focus on the connection between the degree of the
polynomials in a rational function and its horizontal asymptote—thats where the
limit at infinity comes into play.
I. Foundational Concepts and Definitions
An asymptote is a straight line such that the distance between the curve and the
line approaches zero as one or both of the x or y coordinates tend to infinity. In
Math 150A, we classify them into three main types:
1. Vertical Asymptotes (VA)
2. Horizontal Asymptotes (HA)
3. Slant (or Oblique) Asymptotes (SA)
A. Vertical Asymptotes (VA)
A vertical line x=a is a Vertical Asymptote of the function f (x) if at least one of the
following limit statements is true:
lim f (x)=∞or lim f ( x)=− ∞
− − +¿
lim ¿
x→ a x→a x →a f (x)=∞ or lim ¿¿
x→a+ ¿ f (x)=−∞ ¿
Conceptual Understanding: VAs occur where the functions denominator is
zero and the numerator is non-zero. This creates an "infinite discontinuity" or
a blow-up in the graph.
N (x )
Procedure for Finding VAs (for Rational Functions f ( x)= ):
D(x )
a. Factor both the numerator N ( x ) and the denominator D( x )
completely.
, b. Cancel any common factors. (If a factor cancels, it indicates a hole or
removable discontinuity, not a VA.)
c. Set the remaining, simplified denominator equal to zero and solve
for x . These x -values are the locations of the Vertical Asymptotes.
B. Horizontal Asymptotes (HA)
A horizontal line y=L is a Horizontal Asymptote of the function f (x) if:
lim f (x)=Lor lim f (x )=L
x→ ∞ x→ −∞
Conceptual Understanding: HAs describe the end behavior of the function
—what y value the graph settles on as x gets extremely large (positive or
negative). A function can have at most two HAs (one for x → ∞ and one for
x → − ∞, though rational functions usually only have one).
n
an x +…
Procedure for Finding HAs (for Rational Functions f ( x)= m ):
b m x +…
Let n be the degree of the numerator and m be the degree of the
denominator.
a. Case 1: (Degree of Numerator < Degree of Denominator) n< m
The HA is always y=0 (the x -axis).
Why? The denominator grows much faster than the numerator,
forcing the fraction to approach zero.
b. Case 2: (Degree of Numerator = Degree of Denominator) n=m
an
The HA is y= , the ratio of the leading coefficients.
bm
Why? As x → ± ∞, the lower-order terms become insignificant,
n
an x an
and the function behaves like f ( x)≈ m
= .
bm x bm
c. Case 3: (Degree of Numerator > Degree of Denominator) n> m
There is NO Horizontal Asymptote. The function grows
without bound. (It may have a Slant Asymptote if n=m+ 1).
C. Slant (or Oblique) Asymptotes (SA)
A non-horizontal, non-vertical line y=mx+b is a Slant Asymptote of the function
f ( x) if:
(Calculus I)
A CSUN Students Insight on Mastering Asymptotes
Welcome, fellow Math 150A student! Asymptotes are one of the most crucial
concepts youll encounter in Calculus I, as they connect limits (Chapter 2) with
curve sketching (Chapter 4). Dont just memorize the rules; understand why these
lines exist. An asymptote is a limiting behavior—its a line that a function approaches
but never truly touches as its input (x) or output (y) heads towards infinity.
Mastering this topic means mastering limits involving infinity, which is fundamental
to all future calculus courses. Focus on the connection between the degree of the
polynomials in a rational function and its horizontal asymptote—thats where the
limit at infinity comes into play.
I. Foundational Concepts and Definitions
An asymptote is a straight line such that the distance between the curve and the
line approaches zero as one or both of the x or y coordinates tend to infinity. In
Math 150A, we classify them into three main types:
1. Vertical Asymptotes (VA)
2. Horizontal Asymptotes (HA)
3. Slant (or Oblique) Asymptotes (SA)
A. Vertical Asymptotes (VA)
A vertical line x=a is a Vertical Asymptote of the function f (x) if at least one of the
following limit statements is true:
lim f (x)=∞or lim f ( x)=− ∞
− − +¿
lim ¿
x→ a x→a x →a f (x)=∞ or lim ¿¿
x→a+ ¿ f (x)=−∞ ¿
Conceptual Understanding: VAs occur where the functions denominator is
zero and the numerator is non-zero. This creates an "infinite discontinuity" or
a blow-up in the graph.
N (x )
Procedure for Finding VAs (for Rational Functions f ( x)= ):
D(x )
a. Factor both the numerator N ( x ) and the denominator D( x )
completely.
, b. Cancel any common factors. (If a factor cancels, it indicates a hole or
removable discontinuity, not a VA.)
c. Set the remaining, simplified denominator equal to zero and solve
for x . These x -values are the locations of the Vertical Asymptotes.
B. Horizontal Asymptotes (HA)
A horizontal line y=L is a Horizontal Asymptote of the function f (x) if:
lim f (x)=Lor lim f (x )=L
x→ ∞ x→ −∞
Conceptual Understanding: HAs describe the end behavior of the function
—what y value the graph settles on as x gets extremely large (positive or
negative). A function can have at most two HAs (one for x → ∞ and one for
x → − ∞, though rational functions usually only have one).
n
an x +…
Procedure for Finding HAs (for Rational Functions f ( x)= m ):
b m x +…
Let n be the degree of the numerator and m be the degree of the
denominator.
a. Case 1: (Degree of Numerator < Degree of Denominator) n< m
The HA is always y=0 (the x -axis).
Why? The denominator grows much faster than the numerator,
forcing the fraction to approach zero.
b. Case 2: (Degree of Numerator = Degree of Denominator) n=m
an
The HA is y= , the ratio of the leading coefficients.
bm
Why? As x → ± ∞, the lower-order terms become insignificant,
n
an x an
and the function behaves like f ( x)≈ m
= .
bm x bm
c. Case 3: (Degree of Numerator > Degree of Denominator) n> m
There is NO Horizontal Asymptote. The function grows
without bound. (It may have a Slant Asymptote if n=m+ 1).
C. Slant (or Oblique) Asymptotes (SA)
A non-horizontal, non-vertical line y=mx+b is a Slant Asymptote of the function
f ( x) if:
