Limits, Real Numbers, and Vertical Asymptotes — Comprehensive Notes

Real numbers, imaginary numbers, and infinity

Real numbers: all numbers you typically encounter, including natural numbers (1, 2, 3, …), integers, fractions, decimals, and numbers with infinite decimals like $\pi$. In short, the real numbers $\mathbb{R}$ include counting numbers, negatives, rationals, and irrationals.
Imaginary numbers: numbers that involve $i$ (e.g., $bi$ with $b \in \mathbb{R}$). These are not real numbers and typically are not plotted on the same real number line grids used in introductory calculus.
Infinity and negative infinity: not real numbers. We do not treat $\infty$ or $-\infty$ as actual inputs/outputs for a function; they describe unbounded behavior. In this course, infinity is not treated as a real number; sometimes advanced math uses extended notions, but this class does not.
Why this matters for limits: when we say a limit does not exist because the sided limits are not the same real number, we are usually comparing finite real values. A limit that tends to $\infty$ or $-\infty$ is treated as diverging and does not exist as a finite real number.
Note on rules: math sometimes explores ideas by extending or bending rules (e.g., treating $\infty$ as a number in some contexts). Here, we stay within the standard rule: limits are real numbers if finite; otherwise we say the limit does not exist (diverges).

Limits: definitions and notations

a) Two-sided limit and one-sided limits

Two-sided limit: \lim_{x\to a} f(x) = L means as $x$ gets arbitrarily close to $a$ (but $x \neq a$), $f(x)$ gets arbitrarily close to $L$.
One-sided limits:
- Right-hand limit: \lim_{x\to a^{+}} f(x) = L^{+}
- Left-hand limit: \lim_{x\to a^{-}} f(x) = L^{-}
Existence criterion: a finite two-sided limit $\lim_{x\to a} f(x)=L$ exists iff both one-sided limits exist and are equal to the same real number $L$.
If either one-sided limit is $\pm\infty$, the two-sided limit does not exist as a real number (the function diverges at $a$).
If the one-sided limits exist but are different, the two-sided limit does not exist.
Important distinction: a limit can exist (finite) even if $f(a)$ is not equal to the limit or even if $f(a)$ is undefined (hole). In that case the limit exists, but the function value at $a$ may be discontinuous or undefined.

b) Examples from the transcript

Example 1: lim as $x \to -3$ does not exist because the right-hand limit tends to $+\infty$ (diverges). This violates the requirement for a finite real limit.
- Right-hand: $\lim_{x\to -3^{+}} f(x) = +\infty$ (diverges).
- Therefore $\lim_{x\to -3} f(x)$ does not exist as a finite real number.
Example 2 (graph 2): there are many limits for different $x$-values; for instance, \lim_{x\to 5} f(x) = 8.
Example 3: at $x=3$, the right-hand limit is $4$ and the left-hand limit is $1$; since $4 \neq 1$, \lim_{x\to 3} f(x) does not exist.
Example 4: at $x=0$ in graph 3, the limit does not exist because as you approach $0$ from the right you go to $-\infty$ and as you approach from the left you also go to $-\infty$; since $-\infty$ is not a real number and the two one-sided limits are not equal as real numbers, the two-sided limit does not exist (in the real-number sense). In extended-reals this could be written as $\lim_{x\to 0} f(x) = -\infty$, but the transcript treats it as not existing as a finite real.

c) Vertical asymptotes and their link to limits

Vertical asymptote definition (as described): if \lim_{x\to a} f(x) = \pm\infty, then the line $x=a$ is a vertical asymptote of the graph of $y=f(x)$.
Intuition: the function grows without bound as $x$ approaches $a$ from one or both sides, indicating a “blow-up” in the graph near that $x$-value.
In the activity, students were asked to identify which graphs exhibit the property that $f(x) \to \pm\infty$ as $x \to a$ (from either side). The circles mark vertical asymptotes where the limit diverges to $\pm\infty$.

How many limits exist and where they do not exist

A function typically has many limits at different input values. To specify where a limit does not exist, you must state the input value $a$.
Examples from discussion:
- At $x=-3$ and $x=3$ there are limits that do not exist (for different reasons: divergence to infinity or mismatched one-sided limits).
- At $x=0$ in one graph, the limit does not exist because both sides diverge to negative infinity (a vertical asymptote), so the limit is not a finite real number.
A limit can fail to exist for two broad reasons:
- One-sided limits diverge to infinity or negative infinity, or are not equal to each other.
- The left and right finite limits exist but are not equal.

A guided sketch problem: combining several limit properties

Target properties to realize in a single function:
- $f(-2) = 2$
- $\displaystyle \lim_{x \to -2} f(x) = 1$ (the limit near $-2$ is 1, but the function value at $-2$ is 2; a removable discontinuity/hole at $(-2,1)$ is allowed).
- $\displaystyle \lim_{x \to 0} f(x) = 3$ and $f(0) = 3$ (continuous at $0$ here).
- $f(1)$ is not defined (hole or vertical asymptote at $x=1$ is possible).
Conceptual construction notes:
- The existence of a hole at $x=-2$ is consistent with $f(-2)=2$ while the nearby outputs approach $1$.
- At $x=0$, since the limit equals the value, you can connect smoothly through $(0,3)$.
- At $x=1$, you can choose either a hole (removable discontinuity) or a vertical asymptote, consistent with the idea that $f(1)$ is not defined.
The rest of the graph is flexible; many different graphs can satisfy these properties.
Takeaway: limits give strong information about the graph’s shape and discontinuities; direct substitution is possible in the right circumstances (see below).

Direct substitution: when it works and when it doesn’t

The idea behind direct substitution is intuitive: if the function is well-behaved (continuous) at $a$, then you can often compute the limit by evaluating the function at $a$:
- If $\lim_{x\to a} f(x) = f(a)$, then you can substitute $x=a$ to find the limit.
Scenarios where direct substitution does not work (or may mislead):
- If there is a vertical asymptote at $a$ (the limit diverges), substitution would try to compute something like a division by zero, which is undefined.
- If there is a removable discontinuity at $a$ (a hole), the function value at $a$ may be undefined or different from the limit; however, the limit may still exist (e.g., $\lim_{x\to a} f(x) = L$ even if $f(a)$ is not defined or $f(a) \neq L$).
Examples from the discussion:
- Graph 2 (second graph): approaching $x=3$, the left and right limits may differ or involve a hole; direct substitution at $x=3$ would not reflect the actual limit if there is a hole or a discontinuity; in the transcript, the described situation yields a limit of 6 even though a hole is present at $x=3$.
- Graph 2 (alternative subcase): if the denominator tends to $0$ as $x\to 3$ (e.g., a factor $x-3$ in the denominator) and the graph shows a vertical asymptote, the limit does not exist due to opposite or unbounded behavior from sides.
Practical heuristic:
- If the point $x=a$ is in the domain and the function is continuous there, then direct substitution works: \lim_{x\to a} f(x) = f(a).
- If $x=a$ is not in the domain but the limit exists (removable discontinuity), the limit exists but you cannot simply substitute to get a finite value unless you consider the limit value $L$ separately.
- If $x=a$ is a vertical asymptote, or the left- and right-hand limits are different, the limit does not exist as a real number.
A practical note from the class discussion: there are at least two cases where substitution is straightforward and useful:
- When $a$ is a point where the function is defined and continuous there.
- When the limit exists at $a$ and the function value at $a$ matches that limit (i.e., no discontinuity at $a$).
A reminder about help resources: if you’re unsure about a limit or how to substitute, discuss with instructors or office hours for guidance.

Worked ideas and takeaways

Limits are about the behavior of f(x) as x gets close to a, not necessarily about the value at a.
Vertical asymptotes are closely tied to limits that diverge to infinity; they indicate that the graph shoots off to unbounded values near some x-value.
The same function can have multiple limits at different x-values; you must specify which input value you’re considering when stating where limits do not exist.
Holes vs vertical asymptotes:
- Hole (removable discontinuity): the function is not defined at a value, but the limit exists and equals some finite value. You can sketch a hole at that x-value and a y-value equal to the limit.
- Vertical asymptote: the function grows without bound as x approaches the value from one or both sides; you may see a hole or just a vertical line with unbounded behavior, depending on the function.

Quick practice-style recap

Identify where limits do not exist by inspecting one-sided limits:
- If $\lim_{x\to a^{+}} f(x) = +\infty$ or $-\infty$ (or any unsymmetric infinite behavior), the two-sided limit does not exist.
- If $\lim{x\to a^{+}} f(x)$ and $\lim{x\to a^{-}} f(x)$ exist but are not equal, the two-sided limit does not exist.
Use vertical asymptote criterion to flag potential non-existence: if a limit tends to $\pm\infty$ as $x\to a$, you have a vertical asymptote at $x=a$.
Use direct substitution when appropriate: if the limit exists and equals the function value at that point, you can substitute to compute it; if not, consider the left/right limits or the possibility of a hole/vertical asymptote.

Notation quick reference

Two-sided limit: \lim_{x\to a} f(x) = L
One-sided limits:
- Right: \lim_{x\to a^{+}} f(x) = L^{+}
- Left: \lim_{x\to a^{-}} f(x) = L^{-}
Divergence to infinity: \lim_{x\to a^{+}} f(x) = +\infty\quad\text{or}\quad -\infty

- Vertical asymptote at $x=a$ if \lim_{x\to a} f(x) = \pm\infty

emovable discontinuity (hole): a point $x=a$ where $f(a)$ is undefined (or different from the limit) but \lim_{x\to a} f(x) = L exists

Connections to broader ideas

Limits underpin instantaneous rate of change (the derivative) and continuity.
Understanding when limits exist and when they diverge helps explain graph shapes (peaks, holes, asymptotes) and informs algebraic techniques for evaluating limits.
Real-world relevance: limits describe behavior near points of interest, such as approaching a boundary, approaching a critical point in a model, or understanding asymptotic behavior in applied problems.

Ethical, philosophical, and practical implications

Math rules are often guidance for precise reasoning; sometimes advanced contexts extend or reinterpret rules (e.g., working with infinity as an extended value). In learning contexts, follow the standard definitions unless explicitly introducing an extended framework.
Visual intuition from graphs is powerful, but formal definitions ensure consistent conclusions, especially near discontinuities or unbounded behavior.