Limits, Graphs, and Asymptotes (G(x) and g(x))

Topic: Limits, one-sided limits, and asymptotes from graphical contexts (AP Calculus AB, Test #1 materials)
Two graphs are referenced: G(x) on page 1 and g(x) on page 2 (Handout #1: Limits: A Graphical Approach).
Key concepts covered by the transcript:
- Existence vs. nonexistence of limits
- Infinite limits and vertical asymptotes (VA)
- Horizontal asymptotes (HA) and end behavior
- One-sided limits (e.g., x
$\to$ a+, x
$\to$ a
$\text{--}$ )
- Discontinuities and function values at points of discontinuity (e.g., g(a) undefined but limit may exist)
- Use of the graph to determine limits at specific x-values and at infinity
Specific observed ideas from G(x) (as described in the transcript):
- There are potential vertical asymptotes at x = a, x = b, and x = c (as the transcript mentions x
$\text{--}$ a, x
$\text{--}$ b, x
$\text{--}$ c and asks to explain VA at these points).
- There is a horizontal asymptote at y = 0 (the line y = 0 is described as a HA for G(x);
  the transcript even states: "Explain why y = 0 is a horizontal asymptote to G(x)."; it also notes f(x) = 0 and thus Y = 0 as HA).
- Behavior near VA:
- As x approaches a from the right (x
$\to$ a+), G(x) tends to $\infty$ (lim G(x) = $\infty$ as x
$\to$ a+).
- There are one-sided limit indications around a and b (x
$\text{--}$ a, x
$\text{--}$ b with +/
$\text{--}$ superscripts) suggesting different one-sided behaviors near those VA points.
- Some limits listed in the transcript are $\infty$ , 0, or DNE (does not exist). Specific values depend on the graph, but the key takeaway is to identify:
- when a limit blows up to $\infty$ or $\text{--}\infty$ , you have a vertical asymptote at the corresponding x-value;
- when the function approaches a finite value at infinity, you have a horizontal asymptote at that finite value;
- when one-sided limits disagree or do not exist, the two-sided limit DNE.
- A particular one-sided limit is shown as: lim
_{\text{x\to c+}} G(x) = 0 (according to the line "11. lim G(x) = 0 x
$\text{--}$ c+"); this would indicate a behavior where approaching c from the right yields 0 (though this would not by itself contradict VA unless the graph shows VA from at least one side with unbounded behavior). This highlights the importance of distinguishing VA (unbounded near the point) from a finite one-sided limit.
- The transcript also includes a question prompting whether there is a VA at x = a, with the answer: yes, because lim
_{\text{x\to a}} f(x) = $\,\infty$ , so x = a is a VA. This reinforces the principle that an unbounded limit at a point implies a vertical asymptote there.
- The graph hint given: y = 0 is a HA; x = c is a VA; and x = a is a VA (with right-hand limit $\,\infty$ ).
Handout #1 (Graph of g(x))
- The questions on the graph ask for limits like lim g(x) = h, lim g(x) = i, and other limit values, including limits that equal $\,\infty$ or DNE.
- A notable point from the transcript: 11. g(a) = undefined, which illustrates a removable/discontinuous point where the limit may exist even though the function value is not defined there.
- Other items indicate g(b) = m and g(0) = k, showing that the graph includes specific function values at certain x-points that may or may not align with the limit at those points.
- The presence of 14. lim g(x) = (x
$\text{--}$ 0) suggests a limit tied to a behavior around x = 0 (possibly indicating a simplification or a comparison to a linear behavior near 0).
Core definitions and concepts to remember (with formulas):
- Limit existence and nonexistence:
  - lim
  _{\text{x\to a}} f(x) exists if both one-sided limits exist and are equal: lim
  _{\text{x\to a}} f(x) = L
  - If the two one-sided limits are not equal or do not exist, then lim
  _{\text{x\to a}} f(x) does not exist (DNE).
- One-sided limits:
  - lim
  _{\text{x\to a+}} f(x) = L+ (right-hand limit)
  - lim
  _{\text{x\to a-}} f(x) = L
  $\text{--}$ (left-hand limit)
- Infinite limits and vertical asymptotes:
  - If lim
  _{\text{x\to a}} f(x) = $\,\pm\infty$ , then x = a is a vertical asymptote (VA).
  - Often, at least one one-sided limit tends to $\,\pm\infty$ when x approaches a VA.
- Horizontal asymptotes and end behavior:
  - If lim
  _{\text{x\to\pm\infty}} f(x) = L, then y = L is a horizontal asymptote.
  - The typical scenario in the transcript is y = 0 as HA, i.e., lim
  _{\text{x\to\pm\infty}} G(x) = 0.
- Removable discontinuities:
  - If lim
  _{\text{x\to a}} f(x) exists but f(a) is undefined, the graph has a removable discontinuity at x = a (g(a) = undefined in Handout #1 is an example).
- Example relationships to reinforce concepts:
  - For f(x) =
  1) A product of linear factors in the denominator such as f(x) = $\frac{1}{(x-a)(x-b)(x-c)}$
  would produce vertical asymptotes at x = a, x = b, x = c and, as $|x| \to \infty$ , f(x)
  $\to$ 0, yielding a horizontal asymptote y = 0.
Practical implications and connections:
- Understanding graphically where a function behaves badly (VA) helps in domain specification and in planning limits approaching those points.
- Recognizing when a limit exists but the function value does not is essential for identifying removable discontinuities and for evaluating continuity properties.
- Horizontal asymptotes provide insight into long-run behavior of a model or function, important in applications like physics or economics where predictions for large x are needed.
Summary guidance for exam-style questions:
- Identify all VA candidates by checking if the limit as x approaches a finite value is unbounded ( $\,\pm\infty$ ).
- Identify HA by checking end behavior as x
$\to$ $\,\pm\infty$ .
- Distinguish between lim
\text{x\to a} f(x) and f(a). If f(a) is undefined but the limit exists, note a removable discontinuity.
- Use one-sided limits when the graph indicates different behavior on either side of a point (e.g., x
$\to$ a+ vs x
$\to$ a
$\text{--}$ ).
Connections to broader calculus topics:
- Limits underpin the formal definition of continuity and derivatives.
- Graphical limit reasoning aligns with algebraic limit techniques and helps interpret piecewise functions and rational functions with asymptotes.
Notation recap (LaTeX):
- Horizontal asymptote: y = L if
 $\boxed{ \, \lim_{x\to\pm\infty} f(x) = L \, }$
- Vertical asymptote: x = a if
 $\boxed{ \lim_{x\to a} f(x) = \pm\infty \, }$
- One-sided limits:
 $\lim{x\to a^+} f(x) = L+ , \quad \lim{x\to a^-} f(x) = L-$
Quick takeaway for G(x) from the transcript:
- Vertikalnyye pryamye: x = a, x = b, x = c are indicated as vertical asymptotes (VA).
- G(x) has a horizontal asymptote at y = 0.
- Some one-sided limits approach $\,\infty$ (e.g., as x
$\to$ a+), reinforcing the VA at those x-values.
- The graph includes finite values at certain x (e.g., g(a) = undefined, g(b) = m, g(0) = k) illustrating removable and non-removable discontinuities.
Note: The exact numeric limits (values like h, i, d, etc.) depend on the specific graph being referenced and are not fully legible in the transcript. The structural takeaways (VA, HA, one-sided limits, and discontinuities) are the primary concepts demonstrated by the material.

Key Testable Concepts and Pitfalls

Precise definitions for asymptotes:
- A Vertical Asymptote (VA) occurs at $x = a$ if $\lim_{x\to a} f(x) = \pm\infty$ . It's crucial to understand that even one-sided limits approaching $\pm\infty$ from either side are sufficient to establish a VA.
- A Horizontal Asymptote (HA) occurs at $y = L$ if $\lim_{x\to \pm\infty} f(x) = L$ . This describes the end behavior of the function, not behavior at a finite x-value.
Distinguishing between Limit Value and Function Value:
- A limit $\lim_{x\to a} f(x)$ describes the behavior around $x = a$ , not necessarily at $x = a$ .
- $f(a)$ is the actual value of the function at $x = a$ .
- Removable Discontinuity: Occurs when $\lim{x\to a} f(x)$ exists (a finite value) but $f(a)$ is either undefined or $f(a) \neq \lim{x\to a} f(x)$ .
Existence of a Two-Sided Limit:
- $\lim{x\to a} f(x)$ exists if and only if both one-sided limits exist and are equal: $\lim{x\to a^-} f(x) = \lim_{x\to a^+} f(x) = L$ .
- If one-sided limits are different or one/both do not exist, the two-sided limit DNE (Does Not Exist).
Graphical Interpretation:
- Be prepared to identify VAs by vertical lines where the graph shoots up/down to $\pm\infty$ .
- Identify HAs by observing the graph flattening out horizontally as $x \to \pm\infty$ .
- Locate jump discontinuities (where one-sided limits differ) and removable discontinuities (holes).
- Determine function values at specific points, including when they are undefined or distinct from the limit at that point.
Common Mistakes:
- Conflating the definition of a VA with a jump discontinuity.
- Stating a horizontal asymptote at a finite x-value instead of at infinity.
- Incorrectly evaluating one-sided limits from a graph, especially when there's a hole or a jump.
- Assuming a function is defined at a point just because a limit exists there.