Notes on Limits, Indeterminate Forms, and Infinity from Transcript

Key Concepts and Goals

• Understand and resolve limits that produce indeterminate forms like 0/0 and ∞/∞ by algebraic and trig manipulations, rather than just plugging in.
• Use standard limit facts, identities, and the squeeze theorem to evaluate tricky limits.
• Compare growth of functions at infinity to determine limits based on dominant terms and neglectable (negligible) lower-order terms.
• Translate trigonometric limits into algebraic forms you can manipulate (e.g., write csc x as 1/sin x, use bounds on cosine, etc.).
• Distinguish behavior at infinity for polynomials vs exponential growth; remember that exponential growth dominates polynomial growth.
• Be mindful of sign and domain issues when square roots and absolute values are involved (e.g., sqrt(x^2) = |x|).

Standard limits to keep in memory

• Fundamental limit: \lim_{x\to 0} \frac{\sin x}{x} = 1.
• Reciprocal trig near specific angles: for example, as x approaches (\frac{\pi}{2}), since (\sin x \to 1), \lim{x\to \frac{\pi}{2}} \csc x = \lim{x\to \frac{\pi}{2}} \frac{1}{\sin x} = 1.
• Basic bound for cosine: for all real y, -1 \le \cos y \le 1.

Key technique highlights

• Indeterminate forms 0/0 and ∞/∞: rewrite or manipulate the expression so the indeterminacy is removed (e.g., factor, divide by the highest power, use trigonometric identities).
• Trigonometric manipulation examples:
  • For \lim_{x\to 0} x\bigl(1 - \cos(1/x)\bigr), use that -1 \le \cos(1/x) \le 1 \Rightarrow 0 \le 1 - \cos(1/x) \le 2. Multiplying by x and applying the squeeze theorem (careful with signs of x) yields the limit 0.
  • For limits involving expressions like cos(1/x) with x near 0, you can bound the cosine term and squeeze the rest.
• Squeeze Theorem pattern to remember: if you can bound a function between two other functions that share the same limit L, then your function also tends to L.

Worked example: limit with cosine bound (squeeze)

• Consider \lim_{x\to 0} x\bigl(1 - \cos(1/x)\bigr).
  • Let y = 1/x. Then as x → 0, y → ±∞ is not needed for the bound; we just use the bound on cosine:
  • Since -1 \le \cos y \le 1 for all y, we have 0 \le 1 - \cos y \le 2. Setting y = 1/x gives
  • For x > 0: 0 \le x(1 - \cos(1/x)) \le 2x \to 0 as x → 0.
  • For x < 0: 2x \le x(1 - \cos(1/x)) \le 0 and since 2x → 0 and 0 → 0, by squeeze the limit is 0.
  • Therefore, \lim_{x\to 0} x\bigl(1 - \cos(1/x)\bigr) = 0.
• Takeaways: bound cos; multiply by x; apply squeeze; careful with sign when x passes through 0.

Example: limits at infinity – dominant terms and negligible terms

• Big idea: as x → ∞, the highest-degree terms dominate; lower-degree terms become negligible.

• Illustrative graph intuition: two polynomials like f(x) = 2x^2 + 17x + 39 and g(x) = 2x^2 both go to ∞, but the linear and constant terms become negligible for large x; the leading behavior is governed by 2x^2.

• Exponential functions outrun polynomials: exponential growth dominates polynomial growth for large x.

• Example 1: limit of a rational function where the denominator's degree is higher

  • \lim_{x\to \infty} \frac{5+x}{3+x^2} = 0.
  • Reason: denominator grows like x^2 while the numerator grows like x, so the ratio tends to 0.

• Example 2: limit where degrees are the same in leading terms

  • \lim_{x\to \infty} \frac{3+x^2}{5+x^2} = 1.
  • Reason: dominant terms are both x^2 with coefficients 1; lower terms vanish in the limit, leaving \frac{1}{1} = 1.
  • Note: This corrects a common intuition from transcripts: the limit is 1, not infinity.

• Example 3: constant factors with same leading power

  • \lim_{x\to \infty} \frac{6x^3}{10x^3} = \frac{6}{10} = \frac{3}{5}.
  • After canceling the common highest power, only the constants remain.

• Example 4: square root term and sign behavior

  • \lim_{x\to \infty} \frac{x}{\sqrt{x^2}} = 1, since \sqrt{x^2} = |x| = x) for x → ∞.
  • \lim_{x\to -\infty} \frac{x}{\sqrt{x^2}} = -1, since for x → -∞, |x| = -x.
  • Important nuance: in general, sqrt(x^2) = |x|, not x.

• Example 5: limits involving a logarithm with a rational inner function

  • Consider \lim_{x\to \infty} \left( \frac{5}{2} + \ln\left(\frac{x^2+1}{x^2}\right) \right).
  • Inner ratio: \frac{x^2+1}{x^2} = 1 + \frac{1}{x^2} \to 1.
  • So \ln\left(\frac{x^2+1}{x^2}\right) \to \ln(1) = 0.
  • Therefore the overall limit is \frac{5}{2} + 0 = \frac{5}{2}.
  • Guiding principle: when taking limits of composite expressions, focus on the limit of the inner function (provided it stays within the domain where the outer function is defined), then apply the outer limit accordingly.

• Example 6: attention to “negligible” constants

  • For example, in a ratio like \frac{x^2+3}{x^2+5} (leading terms both x^2), the constants 3 and 5 are negligible for large x, yielding limit 1.

• Example 7: trig identity hint used for an indeterminate form

  • A box problem mentioned: \lim_{\theta\to 0} \frac{1 - \cos^2\theta}{\theta}.
  • Use the identity 1 - \cos^2\theta = \sin^2\theta and the standard limit \lim{\theta\to 0} \frac{\sin\theta}{\theta} = 1. Then the limit becomes \lim{\theta\to 0} \frac{\sin^2\theta}{\theta} = \lim_{\theta\to 0} \left( \frac{\sin\theta}{\theta} \cdot \sin\theta \right) = 1 \cdot 0 = 0.
  • The key move is to rewrite to a form you can apply known limits, then use limit laws with inner function focus.

Practical takeaways for solving limits

• When facing 0/0 or ∞/∞, look for:
  • Algebraic simplifications (factoring, dividing by highest power, factoring x^n, etc.).
  • Trigonometric identities and standard trig limits to rewrite in non-indeterminate forms.
  • The Squeeze Theorem when you can bound an expression between two known limits.
  • Dominant-term analysis at infinity: keep the highest-degree terms; drop negligible constants and lower-degree terms.
  • For expressions of the form ln(h(x)), if h(x) → L > 0, you can often use lim ln(h(x)) = ln(lim h(x)).
• Be mindful of absolute-value and sign issues (e.g., sqrt(x^2) = |x|).
• Don’t rely solely on calculators for limits; being able manipulate and recognize standard forms saves time on exams.

Quick pointers related to the transcript context

• The instructor emphasized memorized identities (e.g., lim x→0 sin x/x = 1, cos bounded by [-1,1]) to quickly resolve limits.
• The squeeze theorem was reinforced as a primary tool for limits like lim_{x→0} x(1 - cos(1/x)).
• Infinity behavior: focus on the most powerful term; continue to treat constants as negligible in the limit.
• In problems with square roots and signs, rewrite using |x| to resolve limits at ±∞.
• Always check the inner function when applying limits to composite expressions involving logs or exponentials.

Quick practice checklist (for exam prep)

• Can you identify whether a limit yields 0/0 or ∞/∞ and select an appropriate manipulation?
• Can you apply the squeeze theorem to a limit like lim_{x→0} x(1 - cos(1/x)) and show the bounds converge to the same limit?
• Do you know and can you apply the standard limits: \lim{x\to 0} \frac{\sin x}{x} = 1 and \lim{x\to \frac{\pi}{2}} \csc x = 1?
• Are you comfortable determining which terms are negligible as x → ∞ and performing the leading-term analysis (e.g., for rational functions and for expressions involving sqrt(x^2))?
• Can you apply the limit law for composite functions like \lim_{x\to\infty} \ln\left(\frac{x^2+1}{x^2}\right)$$ by evaluating the inner limit first?

Notes on a couple of transcript-specific points

• A couple of computed results in the transcript were shown with some misstatements (e.g., a limit that would be 1 rather than infinity for a particular expression). The corrected version above aligns with standard limit rules (e.g., (3+x^2)/(5+x^2) → 1 as x → ∞). Always verify with leading-term reasoning and, if needed, by factoring or dividing numerator and denominator by the highest power of x.
• The core takeaway remains: manipulate to reveal the dominant behavior, then apply limit laws or the squeeze theorem as appropriate.