CSUN Math 150A: Calculus I - Study Note
Key Concept Deep Dive: Limits at Infinity (Horizontal Asymptotes)
Course: Math 150A (Calculus I)
Institution: California State University, Northridge (CSUN)
Topic: Limits at Infinity
Goal: Master the algebraic techniques for finding limits as x approaches plus or minus infinity,
and connect these limits to horizontal asymptotes.
I. Defining the Concept: What Happens Far Out?
The concept of a limit at infinity asks a fundamental question about the behavior of a function: As
the input variable (x) gets arbitrarily large (either positive or negative), what value, if any, does
the output variable (f(x)) approach?
Formal Notation:
Limit as x approaches Positive Infinity:
Limit as x approaches infinity of f(x) = L
This means that as x gets larger and larger (moving far to the right on the graph), the value of f(x)
gets closer and closer to the finite value L.
Limit as x approaches Negative Infinity:
Limit as x approaches negative infinity of f(x) = M
This means that as x decreases without bound (moving far to the left on the graph), the value of
f(x) gets closer and closer to the finite value M.
Geometric Interpretation: Horizontal Asymptotes (HA)
If either of the limits above equals a finite number (L or M), then the line y = L (or y = M) is a
Horizontal Asymptote (HA) of the functions graph.
My Insight (Visualizing HA): The HA is the "boundary" or "horizon" that the functions curve
follows as it extends far away from the origin. A function can have up to two different horizontal
asymptotes (one for the right side, x -> infinity, and one for the left side, x -> negative infinity),
although most rational functions we see in 150A have at most one. Crucially, remember that a
function can cross its horizontal asymptote in the middle of the graph; the asymptote only
describes the functions behavior at the extremes.
II. The Core Algebraic Tool: The Reciprocal Limit Theorem
The entire method for computing limits at infinity for rational functions is built on one simple
rule:
The Reciprocal Limit Theorem:
For any positive rational number r,
Limit as x approaches infinity of 1 / (x power r) = 0
Limit as x approaches negative infinity of 1 / (x power r) = 0
The Logic: If you take 1 and divide it by an enormous number (x power r), the result must be an
extremely small number, tending toward zero. This is the only algebraic tool we need. Everything
else is manipulation to use this theorem.
The Standard Technique: Dividing by the Highest Power
To find the limit of a rational function P(x) / Q(x) as x approaches infinity or negative infinity:
Identify the Highest Power of x in the DENOMINATOR. Let this power be x power n.
Divide Every Term (in both the numerator and the denominator) by x power n.
Simplify the resulting expression.
Apply the Reciprocal Limit Theorem: Replace every term of the form (constant / x power r) with
0.
Evaluate the remaining constant or simplified expression.
Example Application:
Find the Limit as x approaches infinity of (3x squared - x + 1) / (x squared + 5x - 4)
Highest power in the denominator is x squared.
Divide every term by x squared:
(3x squared / x squared) - (x / x squared) + (1 / x squared)
/
(x squared / x squared) + (5x / x squared) - (4 / x squared)
Simplify:
3 - (1 / x) + (1 / x squared)
/
1 + (5 / x) - (4 / x squared)
Apply Reciprocal Limit Theorem (as x -> infinity):
Limit = (3 - 0 + 0) / (1 + 0 - 0) = 3 / 1 = 3
Conclusion: The horizontal asymptote is y = 3.
III. Analyzing Rational Functions by Degree (The Shortcut Rule)
While the division method is the only formal method to show your work, knowing the following
shortcut rule is critical for quickly checking your answers on the exam.
Let P(x) be the numerator of degree n and Q(x) be the denominator of degree m.
Case 1: Degree of Numerator < Degree of Denominator (n < m)
Result: The limit is always 0.
Intuition: The denominator grows much faster than the numerator. Example: (x squared) / (x
cubed). As x -> infinity, this approaches 0.
Horizontal Asymptote: y = 0 (the x-axis).
Case 2: Degree of Numerator = Degree of Denominator (n = m)
Result: The limit is the ratio of the leading coefficients.
Intuition: The highest power terms dominate the entire functions behavior. The ratio of the
coefficients is what remains after dividing by the highest power.
Horizontal Asymptote: y = (a_n / b_m).
Case 3: Degree of Numerator > Degree of Denominator (n > m)
Result: The limit is infinity or negative infinity (it does not exist as a finite number).
Intuition: The numerator grows faster, so the function value shoots up or plunges down.
Horizontal Asymptote: None. (Note: It may have a slant/oblique asymptote, but that’s typically
for Math 150B).
IV. Handling Radicals and Square Roots (The Trickiest Case)
When the function involves radicals (square roots), the dividing process changes slightly, and you
must be extremely careful with signs when x approaches negative infinity.
The Correct Dividing Term:
When dividing terms under a square root, remember that the dividing term must be:
Square root of (x squared)
However, we must account for the absolute value definition:
Square root of (x squared) = x
Therefore:
When x approaches infinity (x > 0): x = x. Divide by x.
When x approaches negative infinity (x < 0): x = -x. Divide by -x.
My Insight (Sign Error Trap): The sign error in the negative infinity case (x -> negative infinity) is
the most common mistake. When you move the dividing term
x
inside the square root, it
becomes
x2
. But when you move
x
out of the square root, you must use
¿x∨¿
, which means
using
− x
when
x
is negative. Failing to use
− x
results in an incorrect sign for the limit.
V. Exponential and Logarithmic Limits
We must remember the relative growth rates of functions as x approaches infinity:
Growth Hierarchy (Slowest to Fastest):
log(x) <<< x power n <<< b power x (where b > 1)
This means:
Limit as x approaches infinity of log(x) / x power n = 0
Limit as x approaches infinity of x power n / b power x = 0 (Exponential growth is always faster
than polynomial growth).
Example: Limit as x approaches infinity of (e power x) / (x power 100) = infinity. (The numerator
dominates)
VI. Critical Analysis of Common Pitfalls (易错点解析)
Sign Error with Radicals (as x -> negative infinity): As mentioned, when dividing by x outside the
square root for the negative infinity limit, you must use the substitution
x=− Squareroot (x2)
for the denominator, to ensure the algebraic cancellation is correct.
The Fix: Always divide by
x
(for
x → ∞
) or
− x
(for
x → − ∞
) outside the radical, and
x2
inside
the radical.
Mixing Up Horizontal and Vertical Asymptotes:
HA (Limits at Infinity): What happens to y as x -> infinity or x -> negative infinity. (Looks at the
ends of the graph).
VA (Infinite Limits): What happens to y as x approaches a finite number a (y -> infinity or y ->
negative infinity). (Looks at breaks/holes in the graph).
The Fix: VAs are where the denominator is zero; HAs are determined by comparing degrees.
Never confuse the two!
Forgetting to Check Both Infinities (Only for Non-Rational Functions):
For standard rational functions (no radicals), the limit as
x → ∞
is usually the same as the limit as
x → − ∞
.
However, for functions with radicals (Square root(x^2)), or non-even/odd powers, the limits can
be different (e.g.,
L ≠ M
). You must calculate both sides independently.
Incorrectly Identifying the Dominant Term in the Denominator:
Students sometimes only look at the highest power inside a radical, or inside a grouped term. You
must identify the term with the overall highest effective degree in the entire denominator.
Treating Absolute Value of x as x for all cases:
Remember: Absolute value of x / x is 1 when x > 0 and -1 when x < 0. This ratio is often the source
of the sign flip between the positive and negative infinity limits.
VII. Study Tips for the CSUN Math 150A Exam
Practice with Radicals: Spend 80% of your time practicing limits with square roots, especially
when
x → − ∞
. This is a guaranteed trick question on the exam.
Use the Shortcut (for checking): Use the degree comparison method (Case 1, 2, 3) to predict the
answer quickly, then use the rigorous "divide by highest power" method to write out the formal
solution. If your formal solution doesnt match the shortcut prediction, you made an algebraic
mistake.
Connect to Graphing: Recognize that every time you successfully calculate a finite limit at infinity,
you have found a crucial piece of information for sketching the curve: the horizontal asymptote.
This connection reinforces the concept.
