S2 calc

Course logistics and assessment flow

• Today is Wednesday; focus is on limits and graph descriptions.
• Quiz schedule and reattempt policy (as described):
  • Tuesday: standard two quiz (standard 2).
  • If you took standard 1 and have a reattempt, you may reattempt standard 1 after finishing standard 2.
  • Next week (recitation): you’ll always have a reattempt opportunity for the previous week if needed.
  • Beyond that, you can schedule reattempts with the instructor.
• Engagement activity: sign-in sheet to receive credit for today’s engagement.
  • You should initial/designate your name for credit.
  • Activity: matching descriptions with graphs. Descriptions and graphs do not have a strict one-to-one correspondence; a single graph can match multiple descriptions, and some descriptions may not match any graph. Every graph has at least one matching description.
• In-class process:
  • Students in groups read the descriptions and inspect the graphs to determine which graphs the descriptions describe.
  • A volunteer on each side of the room shares a rough idea first (not required to be a formal solution).
  • Goal: to collectively map descriptions to the appropriate graphs.
  • If your group needs extra paper to work through the matching, raise your hand for a substitute sheet.
• Prior progress recap (from last session):
  • The group discussed several graphs and descriptions (letters such as f, k, l, h, g, j, etc.).
  • The group noted that several graphs have multiple candidate descriptions and that some descriptions may not correspond to any graph.
  • There was some uncertainty about which specific letters corresponded to which graphs; discussion encouraged private think time and then sharing with the class.
• The instructor emphasized that some statements involve approaching certain x-values from the left or from the right and that limits should be written to reflect those one-sided approaches.

Core concepts in limits (key ideas from today)

• Limits describe the trend of function outputs around an input, not necessarily at the input itself.
  • Intuition: as x gets close to a, f(x) gets close to some value (the limit), regardless of what f(a) is.
• One-sided limits and notations:
  • Left-hand limit: \lim_{x \to a^-} f(x)
  • Right-hand limit: \lim_{x \to a^+} f(x)
  • Two-sided limit (if both exist and are equal): \lim_{x \to a} f(x) = L
  • If the left-hand and right-hand limits are not the same, the two-sided limit does not exist:
    \lim{x \to a} f(x) \text{ does not exist if } \lim{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x).
• Encoding descriptions with limits (examples discussed):
  • If a description says the outputs approach a certain value as x approaches a from the left, write the left-hand limit accordingly.
  • If a description refers to approaching from the right, use the right-hand limit notation.
  • If the description refers to approaching infinity in the y-direction, this corresponds to the limit being infinite (positive or negative) as x approaches the given x-value from the specified side.
• End behavior and horizontal asymptotes:
  • End behavior refers to what happens as x grows without bound (
    \lim{x \to \infty} f(x) and \lim{x \to -\infty} f(x)).
  • Horizontal asymptote y = L exists if either of these limits equals a finite number L:
    \lim{x \to \infty} f(x) = L \quad \text{or} \quad \lim{x \to -\infty} f(x) = L \implies y = L \text{ is a horizontal asymptote}.
  • If \lim_{x \to \infty} f(x) = \infty or -\infty, the graph’s end behavior heads upward or downward, respectively.
• Non-existent limits and examples:
  • A limit does not exist when the two one-sided limits are not the same. Example: left-hand limit = 1, right-hand limit = 4 => \lim_{x \to a} f(x) \text{ does not exist}.
  • Some graphs may show that both one-sided limits tend to infinity or negative infinity and thus the limit does not exist as a finite real number; these are not real numbers and do not yield a finite L.
• Important subtleties about graphs and descriptions:
  • A graph may show an arrow pointing toward a value on the left (or right) indicating the limit, even if the function value at the point is different or undefined.
  • The existence of a limit says nothing about the actual value of the function at the exact input; the function value might be open, closed, or undefined at that input while the limit still exists.
  • Descriptions may refer to approaching a value from the left or right, or approaching infinity, or approaching a finite L as x tends to ±∞.

Notation and how to translate descriptions into limits (practical guidance)

• Standard notation for approach from the right/left:
  • Right-hand approach: \lim_{x \to a^+} f(x)
  • Left-hand approach: \lim_{x \to a^-} f(x)
• Translation example from today’s discussion:
  • h describes: "the values of f(x) can be made as close to 4 as we like by taking x sufficiently close to 3 from the left":
    \lim_{x \to 3^-} f(x) = 4.
  • c describes: "the values of f(x) can be made as close to 1 as we like by taking x close to 1" (two-sided):
    \lim_{x \to 1} f(x) = 1.
  • A description like: "the values of f(x) can be made sufficiently large as x approaches 3 from the right":
    \lim_{x \to 3^+} f(x) = +\infty.
  • A description like: "the values of f(x) can be made very large in the negative direction as x approaches 3 from the right":
    \lim_{x \to 3^+} f(x) = -\infty.
• How to express end behavior with infinity:
  • If as x → ∞, f(x) → L (finite): horizontal asymptote at y = L.
  • If as x → ∞, f(x) → ∞ or -∞: end behavior heads up or down; graph extends without bound in that direction.
• Non-existent limits and a concrete example from today:
  • If the left-hand limit and right-hand limit at a are different, then:
    \lim_{x \to a} f(x) \text{ does not exist}.
  • Example discussion: graph 6 has left-hand limit = 1 and right-hand limit = 4, so the two-sided limit does not exist:
    \lim{x \to 3} f(x) \text{ does not exist because } \lim{x \to 3^-} f(x) = 1 \neq \lim_{x \to 3^+} f(x) = 4.

Specific examples and their interpretations (as discussed in class)

• Example: h
  • Description: values of f(x) approach 4 as x approaches 3 from the left.
  • Notation: \lim_{x \to 3^-} f(x) = 4.
  • Graph interpretation: outputs near x = 3 coming from the left are near 4; the exact value at x = 3 may be different.
• Example: c
  • Description: values of f(x) can be made as close to 1 as we like by taking x close to 1.
  • Notation: \lim_{x \to 1} f(x) = 1.
  • Graph interpretation: as x is near 1 (both sides), f(x) is near 1.
• End behavior example (infinite outputs as x grows):
  • Description: as x → ∞, f(x) → -∞ (or +∞, depending on the graph).
  • Notation: \lim{x \to \infty} f(x) = -\infty. Or \lim{x \to \infty} f(x) = +\infty.
  • Graph interpretation: the rightmost arm of the graph heads downward (toward negative infinity) or upward (toward positive infinity).
• Example: left-end behavior approaching a finite value as x → -∞:
  • Description example from today: leftward end of the graph flattens toward y = 1 (or another finite y-value).
  • Notation: \lim_{x \to -\infty} f(x) = 1.
• Two-sided limit existence condition:
  • If the left-hand and right-hand limits exist and are equal, then the two-sided limit exists and equals that common value.
  • If either side diverges, or if they diverge to different finite values, the two-sided limit does not exist.
• Interpreting open vs. closed circles at the input point:
  • The existence of the limit does not require the function to take that limit value at the exact input; the circle can be open or closed and the limit remains about the approach, not the exact point.

Non-existent limits: additional focus (practice focus)

• Concept: a limit does not exist when the left-hand and right-hand limits are not the same real number.
  • Example structure to identify: consider the six graphs discussed in class and identify all limits that do not exist, including:
  • Limits that tend to positive or negative infinity (these are not real numbers and do not yield finite limits).
  • Limits where the left-hand limit and right-hand limit are different (e.g., left = 1, right = 4).
• Concrete classroom example mentioned:
  • Graph 1: a limit that does not exist because positive infinity and negative infinity are not real numbers (non-finite limits).
  • Graph 6: the limit as x → 3 does not exist since \lim{x \to 3^-} f(x) = 1 and \lim{x \to 3^+} f(x) = 4.
• Takeaway: practice identifying non-existent limits by examining one-sided limits and end behavior on the provided graphs.

End-of-session reflections and upcoming topics

• The instructor encouraged ongoing discussion and further exploration of limits on Friday.
• Key takeaways to reinforce before the next class:
  • Distinguish between the limit value and the actual function value at a given input.
  • Use one-sided limits to describe behavior near a point and end behavior to describe behavior as x → ±∞.
  • Recognize that horizontal asymptotes arise from finite end-behavior limits, while unbounded end behavior indicates the graph extends indefinitely in that direction.
  • Practice encoding descriptive statements as limit expressions and interpreting graphs to identify which limit (left, right, two-sided, or infinity) applies.

Quick reference cheatsheet (summary)

• Limits describe behavior around a, not necessarily at a:
  • \lim{x \to a} f(x) = L if \lim{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L.
• One-sided limits:
  • Left: \lim_{x \to a^-} f(x)
  • Right: \lim_{x \to a^+} f(x)
• Infinite limits and end behavior:
  • \lim_{x \to \infty} f(x) = L → horizontal asymptote y = L
  • \lim_{x \to \infty} f(x) = \infty or $-\infty → end arm goes up or down
• Non-existence of limit:
  • If left-hand and right-hand limits are not equal, or if either tends to ±∞ in a way that does not yield a finite L, then the (two-sided) limit does not exist.
• Notation recap (approach directions):
  • Right: \lim{x \to a^+} f(x), Left: \lim{x \to a^-} f(x)
• Important nuance: a limit can exist even if f(a) is undefined or if the graph has a hole or an open circle at x = a.
• Practical skill: translate descriptions into limit statements and read limit statements off graphs (including distinguishing between approaching a from left vs right and identifying end behavior).