Mon Aug 31, 2026 Lecture (L04) Stewart Sect. 1.6 Limits Involving Infinity Part 1: Infinite Limits

Overview

Barsamian's Math Videos introduces Stewart Section 1.6 by extending ordinary real-valued limits to describe functions whose values grow without bound near a finite x-value. Through 1/x², cotangent near π, a rational function near 4, and a piecewise absolute-value example near 0, the lecture develops one-sided infinite limits and vertical asymptotes while emphasizing that infinity is shorthand for a trend—not a number—and cannot be used in arithmetic.

Key takeaways

The notation lim(x→a) f(x)=+∞ means function values can exceed any chosen positive bound when x is sufficiently close to the finite input a; it does not mean the limit equals a real number.
A vertical asymptote x=a is identified by one-sided unbounded behavior, and the left and right limits can have different signs.
For cot(x)=cos(x)/sin(x) near π, the numerator stays near −1 while the denominator changes sign across π, producing −∞ from the left and +∞ from the right.
A two-sided limit can fail because its one-sided limits disagree, as with cot(x) at π and the corrected rational function (x−2)(x−1)/(x−4) at 4.
Never infer an infinite limit by calculating 1/0 or −1/0; determine the sign and unbounded trend of nearby function values instead.
For expressions involving |x|, use its piecewise definition on each side: 1/x+1/|x| equals 0 for x<0 and 2/x for x>0, so its one-sided limits at 0 are 0 and +∞.

Chapters

0:00 Course Website, Office Hours, and the First Quiz

Barsamian directs students to the public course webpage linked from Canvas for course information, office location, and office hours.
Regular office hours are Tuesday, Wednesday, and Thursday from 2–4 p.m.; one session is to be made up later in the week.
The upcoming Wednesday quiz has three problems, and Section 1.6 is not included; the instructor says it will be tested the following week.

3:00 Why 1/x² Has No Ordinary Real-Valued Limit at Zero

For x approaching 0 through positive values, 1/x² grows from 1 at x=1 to 4 at x=1/2, 100 at x=0.1, and 1,000,000 at x=0.001.
Under the Section 1.3 definition, a limit must approach a real number, so lim(x→0) 1/x² does not exist as an ordinary finite limit.
The graph has no finite destination point (0,L); instead, its y-values grow positively without bound.

6:30 Distinguishing Infinite Behavior from the Earlier Limit Laws

The limit laws from Sections 1.3–1.4 describe limits that exist as real numbers, not a quotient whose numerator approaches 1 and denominator approaches 0.
As x approaches 0 in 1/x², the x-values approach a finite input while the y-values become arbitrarily large and positive.
The lecture treats this behavior as a reason to expand the definition of limit rather than apply ordinary quotient arithmetic.

9:00 The Meaning of an Infinite Limit and the Symbol ∞

The notation lim(x→a) f(x)=∞ means f(x) can be made arbitrarily large and positive by taking x sufficiently close to a, with x≠a.
Infinity is not a real number or a graph point such as (0,∞); it is shorthand for an unbounded trend in function values.
With this Section 1.6 convention, lim(x→0) 1/x² is described as +∞, even though it is not a finite real-valued limit.

14:00 One-Sided Infinite Limits Define Vertical Asymptotes

Infinite behavior can be positive or negative and can occur from the left or the right of a finite input a.
A vertical asymptote is a vertical line x=a associated with at least one one-sided limit growing to +∞ or −∞.
The lecture distinguishes Section 1.6 infinite limits, where x approaches a finite number, from a later topic involving x approaching infinity.

17:20 Preparing the Cotangent Example with Sine and Cosine

To analyze lim(x→π) cot(x), Barsamian rewrites cot(x) as cos(x)/sin(x).
The argument relies on familiar sine and cosine graphs: cos(x) approaches −1 near π, while sin(x) approaches 0.
Rather than depend on a calculator or memorize the cotangent graph, the method tracks the signs and relative sizes of numerator and denominator.

22:00 Cotangent Approaches −∞ from the Left and +∞ from the Right

For x→π⁻, cos(x) is near −1 and sin(x) is positive but close to 0, so cos(x)/sin(x) becomes arbitrarily large in magnitude and negative: the limit is −∞.
For x→π⁺, cos(x) remains near −1 while sin(x) is negative and close to 0, making the quotient arbitrarily large and positive: the limit is +∞.
The cotangent graph therefore has a vertical asymptote at x=π, but lim(x→π) cot(x) does not exist because the one-sided limits differ.

30:10 Why Limits Cannot Be Evaluated by Dividing by Zero

The lecture rejects replacing a limiting quotient with expressions such as −1/0 or treating 1/0 as infinity; those are not valid arithmetic operations.
A correct infinite-limit result comes from analyzing values near the input, including their signs—not from substituting the input and dividing by zero.
A shortcut that happens to produce the right result for 1/x² is still invalid if it can give the wrong signs or answer for other limits.

35:20 A Rational Function Near 4 Has Opposite One-Sided Limits

The worked example is analyzed using the corrected numerator x²−3x+2=(x−2)(x−1), whose value at x=4 is 6; the board initially shows a minus sign before the instructor changes it to plus.
As x→4⁻, the numerator stays near 6 and x−4 is negative and near 0, so the quotient tends to −∞.
As x→4⁺, the numerator stays near 6 and x−4 is positive and near 0, so the quotient tends to +∞; consequently, the two-sided limit does not exist.

43:00 Piecewise Absolute Value Produces Different Behavior at Zero

The correct piecewise definition is |x|=x for x≥0 and |x|=−x for x<0; the formula used depends on which side of 0 x approaches from.
For x<0, 1/x+1/|x|=1/x−1/x=0, so the left-hand limit at 0 is 0.
For x>0, 1/x+1/|x|=2/x, which tends to +∞ as x→0⁺; the two-sided limit therefore does not exist.
The graph is y=0 on the negative side and rises without bound on the positive side, combining a finite left-hand approach with a vertical-asymptote trend.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Barsamian's Math Videos.