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Notes on Absolute Value Inequalities and Intuitive Limits

Absolute Value Inequalities (recap and conventions)

  • When solving inequalities involving absolute values, remember the core rules:
    • If you have an inequality of the form |A| \le B with B\ge 0, then the solution is the double inequality
      -B \le A \le B.
    • If you have |A| \ge B with B\ge 0, then the solution is the union
      A \le -B \quad \text{or} \quad A \ge B.
    • If B<0, the inequality has no solution (since the absolute value cannot be strictly less than a negative number, and cannot be greater than a negative bound either).
  • Translating to a concrete example from the transcript:
    • Solve |2x-5| \le 1. Here, A = 2x - 5 and B = 1. Apply the rule:
      -1 \le 2x-5 \le 1. Add 5 to all parts: 4 \le 2x \le 6. Divide by 2: 2 \le x \le 3.
    • Therefore the solution interval is [2,3].
  • Alternative viewpoint (two separate inequalities):
    • From |2x-5| \le 1 you can also write the two inequalities
      2x-5 \le 1 \quad\text{and}\quad -(2x-5) \le 1,
      which simplify to the same overall constraint and yield the same intersection: [2,3].
  • Nice “sanity check” point:
    • If you split the solution space into two disjoint regions (e.g., something like x
  • Notational preferences:
    • When listing interval solutions, it’s common to present the smaller endpoint first (e.g., write [2,3] rather than [3,2]).
  • Quick cheat for absolute-value inequalities in practice:
    • If you see something like |f(x)|\le a, convert to -a \le f(x) \le a and solve the resulting double inequality.
    • If you see |f(x)|\ge a, convert to the union of two simple inequalities and solve each side, then take the union of the results.

Intuitive approach to limits (via pictures and motivation)

  • What a limit means (intuition):
    • Limits capture what the function is trying to do “near” a point, even if the function is not defined or has a hole there.
    • Intuition example: consider a cancelled expression like \frac{x^2-25}{x-5}. After canceling, you get the simplified expression x+5. At x=5 this original expression has a hole (0/0), but the limit as x\to 5 is the value the function would take if the hole were closed, namely 10.
    • This illustrates the idea: the limit is about behavior near the point, not necessarily the exact value at the point.
  • Graphical intuition: a hole at x=a may exist even when the function is otherwise well-behaved nearby; the limit as x\to a can be finite even if the function is undefined at a.
  • Two key ideas when reading graphs:
    • The function value at the point can differ from the limit as you approach that point.
    • The limit cares about values of the function as you get arbitrarily close to the point (but not necessarily at the point itself).
  • One- and two-sided limits (notation and idea):
    • Left-hand limit: \displaystyle \lim{x\to a^-} f(x)\equiv L-
    • Right-hand limit: \displaystyle \lim{x\to a^+} f(x)\equiv L+
    • Two-sided limit exists iff both one-sided limits exist and are equal: \displaystyle \lim{x\to a} f(x) = L \;\text{exists if}\; L- = L_+ = L.
  • Application to piecewise functions: if the left-hand and right-hand limits differ, the two-sided limit does not exist. The speaker notes that in such cases, the standard approach (from the left or from the right) helps understand the function’s behavior around the point.
  • Practical takeaway: when solving limits graphically, focus on where the function tends to as you approach a; be mindful of holes, jumps, and asymptotes that affect the limit but not necessarily the function value at the point.

One-sided limits, existence, and piecewise behavior

  • Direction indicators (notation):
    • Left: \lim_{x \to a^-} f(x)
    • Right: \lim_{x \to a^+} f(x)
  • If a function is piecewise, the two one-sided limits can be different, which means the two-sided limit does not exist.
  • If the left-hand limit equals the right-hand limit, the two-sided limit exists and equals that common value.
  • Example discussion from the transcript (conceptual, not a full numeric example):
    • For a piecewise function with a jump at a=1, the left-hand limit might be 2 and the right-hand limit might be -1, so the two-sided limit at 1 does not exist.
  • A caution about algebraic treatment: sometimes textbooks approach these limits by treating the one-sided limits separately (splitting at the point where the behavior changes) to simplify signs and behavior around the point.

Infinite limits and vertical asymptotes

  • When a limit tends to infinity, we write the limit with infinity as the value (not a finite number):
    • \displaystyle \lim_{x\to a} f(x) = \infty or -\infty\, to indicate unbounded growth in that direction.
    • Technically, infinity is not a number; it’s a behavior descriptor. That’s why interval notation uses parentheses around infinity endpoints (e.g., (0, \infty) ) rather than square brackets.
  • Vertical asymptote definition (informal): a vertical line x = a where the function blows up to ±∞ on at least one side as x approaches a. In practice this means at least one one-sided limit is ±∞.
  • Examples from the lecture:
    • 1/x has a vertical asymptote at x = 0: as x → 0^+, f(x) → ∞ and as x → 0^−, f(x) → −∞; the two one-sided limits differ, so the two-sided limit does not exist, and x=0 is a vertical asymptote.
    • 1/x^2 has a vertical asymptote at x = 0 with both one-sided limits → ∞; the two-sided limit is commonly described as ∞ (not a real number).
    • ln(x) has a vertical asymptote at x = 0 with the right-hand limit to −∞ (as x → 0^+), and the left-hand side is not defined (there is no left of zero for ln). This emphasizes that the left-hand limit does not exist in the domain of the function.
  • Tangent function example: tan(x) has vertical asymptotes where cos(x) = 0, i.e., at x = \frac{\pi}{2} + k\pi for integers k. The graph shows alternating infinite left/right limits around those points.
  • Important nuance about infinity: infinity is a useful descriptor but not a finite value; some calculus problems will specifically ask for input like "infinity" or "does not exist" rather than a real number.
  • Practical takeaway: identify vertical asymptotes by looking for directions in which the one-sided limits blow up to ±∞.

Limits with basic operations (sum, product, quotient) and cautions

  • If both limits exist (and are finite), then:
    • \displaystyle \lim{x\to a} [f(x) + g(x)] = \lim{x\to a} f(x) + \lim_{x\to a} g(x)
    • \displaystyle \lim{x\to a} [c\,f(x)] = c \cdot \lim{x\to a} f(x) for any constant c.
    • \displaystyle \lim{x\to a} [f(x) \cdot g(x)] = \left(\lim{x\to a} f(x)\right)\cdot\left(\lim_{x\to a} g(x)\right)
    • If the limit of the denominator exists and is nonzero, then
      \displaystyle \lim{x\to a} \frac{f(x)}{g(x)} = \frac{\lim{x\to a} f(x)}{\lim_{x\to a} g(x)}.
  • Important caveat (the converse is not guaranteed): even if the limit of a sum/product exists, it does not necessarily imply that the individual limits exist. Cancellations can make the combined limit finite while the separate limits fail to exist.
    • Classic counterexample for the non-reverse direction: consider a construction where two oscillating functions cancel in the sum, e.g., let f(x) = sin(1/x) and g(x) = −sin(1/x) for x ≠ 0. Then f(x) + g(x) = 0 for all x ≠ 0, so the limit as x → 0 is 0, but neither lim{x→0} f(x) nor lim{x→0} g(x) exists.
  • While algebraic rules make intuition easier, always check the existence of the component limits when using the sum/product/quotient rules.

Quick reference reminders (from the lecture)

  • Mathematically, infinity is not a number; we use ∞ and −∞ to describe unbounded behavior. Write limits that tend to infinity with the corresponding symbol rather than trying to assign a numeric value.
  • A vertical asymptote is a x-value where at least one one-sided limit is ±∞; the function may oscillate wildly elsewhere but near the asymptote it shoots off to infinity.
  • A limit can exist even if the function is not defined at the point; a typical hole in the graph corresponds to a removable discontinuity where the limit exists but the function value is not equal to the limit.
  • For homework and exams in introductory calculus, expect to input expressions like
    • \infty, \ -\infty, \ \text{or} \ \text{does not exist} when describing limits that diverge or are undefined in the usual sense.

Worked prompts and practice ideas (based on the transcript)

  • Solve the following inequalities (practice these patterns):
    • Solve |A| \le B by converting to -B \le A \le B and solving for x.
    • Solve |A| \ge B by splitting into two regions: A \le -B \quad\text{or}\quad A \ge B and solving for x.
  • Work with a function that has a hole at a; compare the function value f(a) with the limit as x → a: illustrate the difference between f(a) and lim_{x→a} f(x).
  • Consider a piecewise function with a potential jump at a; determine the one-sided limits and conclude whether the two-sided limit exists.
  • Graph-based intuition: identify vertical asymptotes by where the left-hand and right-hand limits diverge to ±∞; practice with familiar functions like \frac{1}{x}, \frac{1}{x^2}, \ln x, \tan x to see the different behaviors on either side of critical points.

Quick connections to broader ideas

  • These concepts lay the groundwork for limits in calculus: understanding when limits exist, how to evaluate them using algebraic manipulation, and how graphs reveal behavior near problematic points (holes, jumps, asymptotes).
  • The discussion of one-sided limits and piecewise behavior ties directly to the formal definition of the limit and to continuity concepts studied in later chapters.
  • The rules for limits of sums, products, and quotients foreshadow more advanced limit manipulations (including cases with indeterminate forms) that appear in later calculus topics.

Summary takeaways for the exam

  • For absolute value inequalities, convert to a double inequality when appropriate and solve; remember the correct intervals and endpoints (closed vs open) based on the inequality type (≤,
  • When approaching limits graphically or intuitively:
    • Distinguish between the actual function value at a point and the limit as you approach that point.
    • Use one-sided limits to diagnose the existence of a two-sided limit.
    • Recognize vertical asymptotes as x-values where one-sided limits blow up to ±∞.
    • Treat ∞ and −∞ as descriptive endpoints, not finite numbers; use appropriate notation and brackets/parentheses conventions.
  • If both f and g have finite limits at a, then you can pass limits through sums, products, and quotients (with the caveat that the denominator’s limit must be nonzero). The converse need not hold due to possible cancellations.

If you want, I can turn this into a printable study sheet with compact examples and a few extra practice problems. Also I can add more explicit worked steps for the piecewise limit examples if you’d like to see them written out in full detail.