Wed Aug 26, 2026 Lecture (L02) Stewart Section 1.4 (Calculating Limits)

Overview

Barsamian's Math Videos explains Stewart Section 1.4 by using limit laws to calculate limits and showing when direct substitution is valid. Examples include evaluating a rational function at x = 2, resolving a 0/0 form at x = 3 by factoring and canceling for a limit of 3/5, and simplifying two further indeterminate limits to 12 and -1/49.

Key takeaways

Direct substitution evaluates a polynomial limit at any input and a rational-function limit only when the input is in its domain.
The quotient limit law requires the denominator's limit to be nonzero; if substitution gives 0/0, that law is unavailable.
A 0/0 form is indeterminate rather than a conclusion that the limit is zero or undefined; further algebra may reveal the limit.
Canceling a factor in a limit is justified when the input approaches a value but is not equal to it, so the factor is nonzero for the nearby inputs.
A function and its canceled expression may have different domains: x(x − 3)/((x + 2)(x − 3)) excludes x = 3, whereas x/(x + 2) does not.
The lecture's three algebraic examples resolve 0/0 forms by simplification, yielding limits of 3/5, 12, and −1/49.

Chapters

0:00 Lecture 2: Course Resources and the Previous Recitation

Barsamian's Math Videos identifies the class as Math 2301, Lecture 2, on August 26, covering Stewart Section 1.4.
The course calendar page links to lecture videos and handouts; students can reach the public page through Canvas or bookmark it.
The recitation review revisits Section 1.3 and interpreting limits from graphs and tables.

2:00 Informal Limits and the Graphical Location (a, L)

The notation lim x→a f(x) = L informally means that as x approaches a without equaling it, f(x) approaches L.
A limit describes the graph heading toward the location (a, L), whether or not the function is defined at x = a.
A recitation table showed y-values approaching 3/5 as x approached 3, motivating analytical methods beyond graphs and estimates.

6:30 Why Limit Laws Replace the Precise Definition in Calculus

The precise definition can justify limits rigorously, but the lecture describes it as too difficult for the current Math 2301 course.
Limit laws are presented as theorems derived from the precise definition and its recurring patterns.
Section 1.4 applies those laws to calculate limits analytically rather than relying on tables or graphs.

10:00 Direct Substitution for a Rational Function at x = 2

The example uses f(x) = (x² − 3x)/(x² − x − 6), with denominator factored as (x + 2)(x − 3).
At x = 2, the numerator is −2 and the denominator is −4, so direct substitution gives the limit 1/2.
The quotient law is valid because the denominator's limit is nonzero; direct substitution also applies because 2 is in the rational function's domain.

15:00 Rational-Function Domains and the Direct Substitution Test

For a rational function, the excluded inputs are the values that make its denominator zero.
In the example, x = −2 and x = 3 are excluded, while x = 2 is allowed.
When the input is in the domain, a polynomial or rational function's limit can be found by plugging in that input.

18:00 At x = 3, the Quotient Law Produces an Indeterminate 0/0

Substituting x = 3 into x² − 3x and x² − x − 6 gives 0/0.
Because the denominator limit is zero, the quotient limit law cannot be used for this calculation.
The 0/0 result does not establish that the limit is undefined; it signals that more work is needed.

22:00 Factoring and Canceling to Find the Limit 3/5

Factoring gives x(x − 3)/((x + 2)(x − 3)), making the source of the 0/0 form visible.
Since x approaches 3 but is not equal to 3, x − 3 is nonzero for the nearby inputs under consideration and can be canceled.
The simplified limit is lim x→3 x/(x + 2), so direct substitution gives 3/5.

26:00 Why Canceling After Substitution Is Invalid

Canceling x − 3 after plugging in x = 3 would mean canceling zero, so the apparently correct result 3/5 would come from invalid steps.
The original rational function excludes x = 3, while the simplified expression x/(x + 2) is defined there; they are not the same function over their full natural domains.
The limit calculation is justified by considering inputs near 3 but unequal to 3, canceling the nonzero factor, and then substituting into the simplified expression.

34:50 Expanding a Difference of Cubes to Evaluate a Limit of 12

For lim h→0 [(-2 + h)³ + 8]/h, direct substitution produces the indeterminate form 0/0.
Expanding (-2 + h)³ gives −8 + 12h − 6h² + h³; the constants cancel with +8.
Factoring h from the remaining numerator allows cancellation for h ≠ 0, leaving 12 − 6h + h², whose limit is 12.

43:00 Combining Fractions to Evaluate a Limit of −1/49

The final example considers lim x→−7 [(1/x) + (1/7)]/(x + 7), which initially yields 0/0.
Combining the numerator fractions gives (x + 7)/(7x); writing the denominator factor separately clarifies the algebra.
Because x approaches −7 without equaling it, x + 7 can be canceled, leaving 1/(7x) and the limit −1/49.
The lecture closes by previewing other indeterminate limits and the squeeze theorem for a later class.

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.