2026 Fall Section 2.3 Limits of Functions at Finite Numbers
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Overview
Professor Ara Araeipour explains limits of functions at finite numbers, emphasizing that a limit depends on nearby function values rather than the value at the target point. The lecture develops numerical, graphical, and algebraic methods, then applies one-sided limits and sign analysis to identify infinite limits and vertical asymptotes.
Key takeaways
- A finite limit depends on function values near x=a, not necessarily on f(a); a function can have a limit even when its value is different or undefined.
- A two-sided limit exists exactly when the left-hand and right-hand limits agree, as the step function h(t) demonstrates with limits 0 and 1 at t=0.
- Factoring x²−1 in (x−1)/(x²−1) reveals a removable hole at x=1 and gives the limit 1/2 after cancellation.
- A vertical asymptote has the form x=a and occurs when at least one relevant one-sided limit grows to positive or negative infinity; ln(x) has one at x=0.
- When a denominator approaches zero, its sign must be checked from the specified side to determine whether the quotient tends to positive or negative infinity.
- Odd and even powers change one-sided sign behavior: near x=5, e^x/(x−5)^3 tends to negative infinity from the left, while e^x/(x−5)^4 tends to positive infinity.
Chapters
- Reviews sequences as functions with domain the natural numbers and limits as descriptions of long-term behavior as n approaches infinity.
- Recaps that 1/n^p approaches 0 for p > 0, while the behavior of r^n depends on the base r.
- Connects convergent geometric sums to a/(1-r) when |r| < 1 and defines horizontal asymptotes using limits as x approaches positive or negative infinity.
- Defines the limit of f(x) as x approaches a by tracking values near a from both sides.
- Uses three graph cases to show that the limit can equal L whether f(a)=L, f(a) is a different value, or f(a) is undefined.
- Establishes that a two-sided limit exists only when the left-hand and right-hand limits agree.
- Introduces vertical asymptotes at x=a when a two-sided or one-sided limit grows to positive or negative infinity.
- Examines the limit as x approaches 1 even though the original function is undefined at x=1 and also at x=−1.
- Uses values approaching 1 from below and above, such as 0.99 and 1.01, to estimate a limit of 0.5.
- Interprets the graph as a removable hole at y=0.5 rather than a vertical asymptote.
- Factors x²−1 into (x−1)(x+1), cancels the common factor, and evaluates 1/(x+1) at x=1 to prove the limit is 1/2.
- Defines h(t)=0 for t<0 and h(t)=1 for t≥0, so the value at zero is h(0)=1.
- Finds the left-hand limit at zero as 0 and the right-hand limit as 1.
- Concludes that the two-sided limit at zero does not exist because the one-sided limits disagree.
- At x=2, one graph approaches y=3 from the left and y=1 from the right, so its two-sided limit does not exist.
- At x=5, both sides approach y=2 even though the solid function value is about 1.4, illustrating that the limit can differ from g(5).
- Distinguishes open circles, which indicate missing function values, from solid points, which give values such as f(2)=3.
- Separates a jump at x=2 from a removable hole at x=4, where both sides approach y=4 but f(4) is undefined.
- Extends one-sided limit notation to cases where function values grow without bound toward positive or negative infinity.
- Uses ln(x) as x approaches 0 from the right: the function decreases without bound, making x=0 a vertical asymptote.
- Reads a graph’s behavior near x=2, x=5, and x=−3 to identify vertical asymptotes at all three lines.
- Shows that the left- and right-hand limits at x=−3 can have opposite signs while still making x=−3 a vertical asymptote.
- Explains that direct substitution producing 0/0 is indeterminate and calls for further algebra rather than an immediate answer.
- Rationalizes a numerator involving √(x+4)−2 by multiplying by its conjugate, √(x+4)+2.
- Uses the difference-of-squares identity to remove the radical expression and the zero-producing factor.
- Evaluates the simplified expression at the target value to obtain 1/4, illustrating that 0/0 requires analysis.
- For a numerator approaching −1 over a denominator approaching zero near x=−3, checks the denominator’s sign on each side to distinguish negative infinity from positive infinity.
- Evaluates e^x/(x−5)^3 as x approaches 5 from the left: e^5 stays positive while the odd power makes the denominator negative, so the limit is negative infinity.
- Changes the denominator power from 3 to 4 to show that an even power makes the near-zero denominator positive, giving positive infinity.
- Factors (x²−2x)/(x²−4x+4) into x(x−2)/(x−2)², then finds the limit as x approaches 2 from the left is negative infinity.
Transcript excerpt
0:08 Good morning class. Here's chapter 2, limits section 2.3. Limits of functions at finite numbers. Limits of functions at finite numbers. So we covered limits of sequences, limits of functions at infinity, and now limits of functions at finite numbers. So here's the summary of what we've already covered. Uh let's quickly go over that. A sequence is an ordered list of numbers or function f whose domain is the set of natural numbers 1 2 3 to infinity.
0:56 Limit refers to the long-term behavior of a sequence i.e. what happens to a subn of a sequence in a long limit of a subn as n approaches infinity is l. You say that a subn converges to l limit of a subn as n approaches infinity is infinity or negative infinity or dn does not exist. We say the sequence diverges. To find limit simply plug in and observe couple of important theorems. The limit of 1 / n^ of p as n approaches infinity is zero for p larger than zero. The limit of r^ n as n approaches infinity depends on the base r. If it's between negative 1 and 1 is
1:43 zero. If it's one is one and if it's larger than one it's infinity. geometric sequence when we add the first n terms we get a * 1 - r ^ n / 1 - r. Now if we take a limit of this as n goes to infinity that would be the infinite sum. Now if r is between -1 and one according to this theorem r the^ of n becomes zero as n goes to infinity. So this goes to zero we get a / 1 - r for a geometric sequence b subn + r b subn the solution is b subn is b sub0 or to the power of n and by
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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Ara Araeipour.