Calculus 1000: One sided limits- (Sec 007, September 25)

Overview

Asghar Ghorbanpour defines right- and left-hand limits and explains how to use them to analyze piecewise functions: select the expression valid on the side of approach, then compare the two one-sided limits to determine whether a two-sided limit exists. Examples include a polynomial, |x|/x at 0, and a piecewise function whose behavior at −1 and 0 includes unequal one-sided limits and the oscillating function sin(1/x); the planned discussion of the indeterminate form 0/0 is deferred.

Key takeaways

A right-hand limit uses x-values greater than the target, while a left-hand limit uses x-values less than it; the function value at the target itself does not determine either limit.
For a piecewise function at a branch boundary, choose the formula valid on the side of approach before applying limit laws or direct substitution.
A two-sided limit exists if and only if both one-sided limits exist and are equal.
For |x|/x at 0, the left-hand limit is −1 and the right-hand limit is 1, demonstrating that unequal one-sided limits make the two-sided limit nonexistent.
At 0, sin(1/x) has no right-hand limit because it oscillates, so the associated two-sided piecewise-function limit cannot exist even if the other side has a finite limit.
At an interior point such as −1/2, where one formula applies throughout a neighborhood, evaluate that branch directly rather than unnecessarily splitting into one-sided cases.

Chapters

0:00 Class Agenda and Recording Notes

Asghar Ghorbanpour introduces one-sided limits as the main topic for the September 25 Calculus 1000 session.
The planned second topic is the indeterminate form 0/0, but Ghorbanpour says the class may not reach it.
Ghorbanpour is recording the lecture and will check whether the cameras capture students before deciding whether to post it.

3:00 Right-Hand and Left-Hand Limit Notation

A right-hand limit at a considers values of x greater than a as x approaches a; its notation places a plus sign after a.
A left-hand limit considers values of x less than a; its notation places a minus sign after a.
In both definitions, the limit concerns nearby function values, not the function's value at the point itself.

10:00 Why One-Sided Limits Matter for Piecewise Functions

Ghorbanpour identifies piecewise-defined functions as the main reason to study one-sided limits in this course.
The examples ask students to find left- and right-hand limits for a polynomial at x = 1, |x|/x at x = 0, and functions shown by a graph or piecewise rule.
For a graph-based limit, students should follow only the portion of the graph on the side from which x approaches the target.

14:00 Choosing the Correct Piece When x Approaches 1

For the piecewise function with branch 2x − 1 when x > 1, the right-hand limit at 1 uses that branch.
The branch expression 2x − 1 has limit 1 at x = 1, found by direct substitution into the polynomial.
The left-hand limit must use the branch defined for x < 1; the branch used can differ even though both limits approach the same x-value.

23:00 The Two-Sided Limit Theorem Connects Both Sides

If both one-sided limits exist and are equal, the two-sided limit exists and equals their common value.
Conversely, if the two-sided limit exists, both one-sided limits exist and equal it.
Because polynomials such as 2x − 1 have two-sided limits, their one-sided limits can be evaluated using the same limit value.

33:00 Examples: Matching and Mismatching One-Sided Limits

For f(x) = x² + 3x − 2 at x = 1, direct substitution gives the two-sided limit 2, so both one-sided limits are 2.
For |x|/x at x = 0, the left-hand limit is −1 and the right-hand limit is 1, so the two-sided limit does not exist.
A piecewise example at x = 1 has matching one-sided limits of 1, illustrating how equal side limits establish the two-sided limit.

35:00 Limit Laws Also Apply to One-Sided Limits

Ghorbanpour states that the usual limit laws from the previous class also work for left- and right-hand limits.
Students can apply those laws while preserving the direction of approach, such as x → a⁻ or x → a⁺.
A new exercise asks for two-sided limits at −1, 0, and −1/2 for a piecewise-defined function.

38:00 At x = −1, Compare the Piecewise Branches

For x → −1⁻, the relevant branch is the polynomial x + 2, whose limit is 1.
For x → −1⁺, the applicable rational branch is (x² + 1)/(x − 1), whose limit at −1 is −1.
Since the one-sided limits are 1 and −1, the two-sided limit at −1 does not exist.

44:00 At x = 0, the Oscillation of sin(1/x) Prevents a Limit

Approaching 0 from the left uses the rational branch (x² + 1)/(x − 1), which tends to −1.
Approaching 0 from the right uses sin(1/x), an oscillating function that has no right-hand limit at 0.
Because one one-sided limit does not exist, the two-sided limit at 0 does not exist; evaluating sin(1/x) by direct substitution is not valid.

48:00 At x = −1/2, Stay Within a Single Branch

The point −1/2 lies between −1 and 0, inside the interval where the same rational branch applies on both sides.
For this limit, students can evaluate the applicable expression directly rather than split the work into separate piecewise branches.
Ghorbanpour emphasizes that a piecewise function does not require one-sided analysis at every point—only where the relevant branches change.

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