Calculus 1000-Horizontal asymptotes example & introduction to continuity (Sec 007, October 2, 2026)

Overview

Asghar Ghorbanpour reviews horizontal asymptotes by evaluating limits as x approaches positive and negative infinity, then applies dominant-term factoring to functions involving radicals and exponentials. He introduces continuity through the conditions that f(a) is defined and lim(x→a) f(x) = f(a), using a graph to distinguish removable, infinite, and jump discontinuities at selected points.

Key takeaways

A horizontal asymptote must be determined separately for x→+∞ and x→−∞; a function can have up to two, such as y = 1 and y = −1 in the radical example.
The identity √(x²) = |x| is essential for asymptote calculations because |x|/x approaches 1 at positive infinity and −1 at negative infinity.
For exponential ratios, the dominant base depends on the direction of the limit: the smaller base dominates as x→−∞, while the larger base dominates as x→+∞.
Continuity at x = a requires both that f(a) exists and that lim(x→a) f(x) equals f(a); existence of the limit alone is insufficient.
A removable discontinuity can be fixed by redefining one function value, whereas infinite discontinuities involve unbounded limits and jump discontinuities involve unequal one-sided limits.
The graph examples classify x = 0 and x = 5 as removable discontinuities, x = 2 as an infinite discontinuity, x = 3 as a jump discontinuity, and x = 1 and x = 4 as continuous points.

Chapters

0:00 Lecture Plan: Horizontal Asymptotes and Continuity

Asghar Ghorbanpour plans to finish a horizontal-asymptote example before beginning the definition and main properties of continuity.
The lesson emphasizes writing complete limit calculations for assignments, midterms, and final exams rather than relying on high-school shortcut recipes.
Continuity will continue in the following Monday lecture before the course moves to differentiation.

2:00 Horizontal Asymptotes Require Two Limits at Infinity

A horizontal asymptote y = L exists when lim(x→+∞) f(x) or lim(x→−∞) f(x) equals the finite number L.
Both limits must be checked because a function can have at most two horizontal asymptotes, one associated with each direction.
The class begins with a radical rational expression and an exponential expression as practice problems.

6:00 Radical Example: Factoring the Dominant Power

For the radical example, the numerator and denominator initially produce an ∞/∞ indeterminate form as x approaches either infinity direction.
Ghorbanpour factors dominant powers, using x² inside the square root and separate dominant factors where needed.
Limit laws reduce terms such as 3/x² to zero, but the square-root factor requires careful treatment of √(x²).

10:00 Why √(x²) Equals |x|, Not x

Ghorbanpour stresses the identity √(x²) = |x), because squaring is not one-to-one over all real numbers.
As x→+∞, |x|/x = 1; as x→−∞, |x|/x = −1.
The radical function therefore approaches different finite values in the two directions, producing horizontal asymptotes y = 1 and y = −1.

14:00 Exponential Example: Dominant Bases at Negative Infinity

For an expression of the form (3^x − 2^x)/(3^x − 4^x), every exponential term approaches 0 as x→−∞, creating a 0/0 indeterminate form.
When x→−∞, the exponential with the smaller base is dominant because it decays toward zero more slowly; Ghorbanpour factors 2^x from the numerator and 3^x from the denominator.
Ratios become (3/2)^x and (4/3)^x, both tending to 0 as x→−∞, while the remaining factor (2/3)^x grows without bound.
The negative sign from the simplified numerator makes the overall limit negative infinity, so there is no horizontal asymptote in the negative direction.

26:00 Exponential Example: Dominant Bases at Positive Infinity

For x→+∞, the largest base dominates: 3^x controls the numerator and 4^x controls the denominator.
Factoring those dominant exponentials produces ratios with bases below 1 that approach zero, leaving a finite limit of 0.
The exponential function therefore has the horizontal asymptote y = 0 in the positive direction only.
Ghorbanpour warns that expressions such as (2/3)^∞ should not be treated as direct substitution; the limit must be justified using exponential-function behavior.

30:00 Transition from Limits to the Definition of Continuity

Limits deliberately ignore the function's value at the point being approached, whereas continuity connects the limit to the actual function value.
A function must be defined at a and in a neighborhood around a before continuity at a can be considered.
The continuity condition is lim(x→a) f(x) = f(a); if either the limit or the function value fails to exist, the function is discontinuous.

34:00 Graph Test for Continuity at x = 0, 1, and 2

At x = 0, the graph has a hole, so f(0) is undefined even though the two-sided limit exists; the function is not continuous.
At x = 1, the graph reaches a defined value and the left- and right-hand limits agree with it, so the function is continuous.
At x = 2, the graph has an infinite behavior associated with a vertical asymptote; the limit is not finite, so continuity fails even if a plotted point is assigned.

40:00 Graph Test for Continuity at x = 3, 4, and 5

At x = 3, the left-hand and right-hand limits exist but approach different values, so the two-sided limit does not exist.
At x = 4, the graph passes through the defined function value without a break, making f continuous there.
At x = 5, the function value and limiting behavior do not match, producing a removable discontinuity.

45:00 Three Discontinuity Types: Removable, Infinite, and Jump

A removable discontinuity occurs when the limit exists but f(a) is undefined or differs from the limit; redefining f(a) to equal the limit can repair the gap.
An infinite discontinuity occurs when at least one one-sided limit becomes ±∞, as with the graph's behavior near x = 2.
A jump discontinuity occurs when both one-sided limits exist but are unequal, as at x = 3.
The lecture concludes by identifying the graph's holes, vertical asymptote, and jump as examples to revisit in the next continuity lecture.

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.