Limits: Two-Sided, One-Sided, Laws, and Delta-Epsilon (Comprehensive Notes)

Two-Sided and One-Sided Limits

• Left-hand limit: $\lim_{x\to a^-} f(x)$
• Right-hand limit: $\lim_{x\to a^+} f(x)$
• Two-sided limit exists if and only if the left-hand and right-hand limits exist and are equal:
  • $\lim<em>{x\to a} f(x) = L$ if $\lim</em>{x\to a^-} f(x) = \lim_{x\to a^+} f(x) = L\, (\text{finite})$
• If the two one-sided limits are not equal or if one does not exist, the two-sided limit does not exist:
  • Example from the transcript: $\lim<em>{x\to -8^-} f(x) = 6\quad\text{and}\quad\lim</em>{x\to -8^+} f(x) = 2$
  • Therefore $\lim_{x\to -8} f(x) \text{ does not exist (DNE)}$
• Infinite limits (not finite):
  • $\lim_{x\to a} f(x) = \infty\quad\text{or}\quad -\infty$
  • Infinity is not a real number and is not “on the path”; both sides can approach the same infinity, but that does not yield a finite limit
• Graphical interpretations:
  • A limit exists when the graph approaches the same y-value from both sides near x = a
  • If the graph heads toward infinity or negative infinity, we classify the limit as infinite (not a finite limit)
  • If the left and right approaches differ, we have a jump or a vertical asymptote; in either case, the limit does not exist as a finite number
• Key intuition: associating the approach from left (negative sign for distances) and from right (positive sign) with the concept of approaching the same output value from both sides

Finite vs Infinite Limits and Infinity

• Finite limit: a real number L such that $\lim_{x\to a} f(x) = L$
• Infinite limit: the function grows without bound as x approaches a; classify as infinite, not a finite limit
• Infinite limits are real in the sense of a limit value classification, but the value ∞ is not a number we can reach or substitute into the function
• Visual cues:
  • Vertical asymptote is where the function blows up to ±∞ as x approaches a from one or both sides
  • If both sides blow up to the same sign, we still classify as an infinite limit, not a finite one

Graphical Intuition and Pathologies

• Jump discontinuity: left-hand limit ≠ right-hand limit; no two-sided limit exists
• Vertical asymptote: the function grows without bound near a; limits from sides may be infinite or may not exist
• Infinite limit: both sides approach the same direction to ∞ or −∞; still not a finite limit
• Excessive oscillation (e.g., near 0, a function like $f(x)=6\cos(1/x)$): no limit because the y-values oscillate without settling
• Practical takeaway: If a graph shows an abrupt jump, vertical asymptote, infinite approach, or uncontrolled oscillation near a, the limit at that point does not exist

Numerical Evaluation of Limits (Left and Right)

• Approach from the left: evaluate at points approaching from the left
  • Examples: $f(a-0.1), f(a-0.01), f(a-0.001),\dots$
• Approach from the right: evaluate at points approaching from the right
  • Examples: $f(a+0.1), f(a+0.01), f(a+0.001),\dots$
• If both sides approach the same value, the limit exists and equals that value; if not, it does not exist
• Worked numerical example from the transcript (illustrative):
  • Consider a limit as x approaches 7, using left-side values like 6.9, 6.99, 6.999, etc., and right-side values like 7.1, 7.01, 7.001, etc.
  • Left-side calculations yielded values converging toward approximately $7.7368$; right-side values were shown to lie between approximately $7.7361$ and $7.7375$, consistent with a finite limit around $\approx 7.7368$
• Analytic (algebraic) evaluation after cancellation also yields the limit value (see below) when a common factor cancels and x is substituted after cancellation

Analytic Techniques: Factoring, Cancellation, and Substitution

• When a limit has a rational expression with a factor that cancels, factor both numerator and denominator to identify removable discontinuities and evaluate at the limit point after cancellation
• Example outline from the transcript (structure rather than exact original function):
  • Factor denominator: $x^2 + 5x - 84 = (x-7)(x+12)$
  • If the numerator contains a factor (x-7) that cancels with the denominator, you can simplify and then substitute $x=7$ in the simplified expression
  • Resulting evaluation yielded approximately $\frac{147}{19} \approx 7.7368$ for the limit
• Takeaway: Algebraic simplification is a powerful method to compute limits when direct substitution yields 0/0 or indeterminate forms

Standard Limits (Four Examples) and Their Values

• Classic limits in radians:
  • $\displaystyle \lim_{x\to 0} \frac{\sin x}{x} = 1$
  • $\displaystyle \lim_{x\to 0} \frac{\cos x - 1}{x} = 0$
  • $\displaystyle \lim_{x\to 0} \frac{e^{x} - 1}{x} = 1$
• Limit at x → 1 for logarithm:
  • $\displaystyle \lim_{x\to 1} \frac{\ln x}{x-1} = 1$
• Numerical thoughts (rigorous details require radians and series expansions):
  • These limits are standard results used to build intuition about the behavior of trigonometric, exponential, and logarithmic functions near 0 or 1
  • In practice, one can verify by series expansions: sin x ~ x, cos x ~ 1 − x^2/2, e^x ~ 1 + x, ln x ~ (x−1) near x = 1
• Note on direction and units:
  • The first two limits require radians; keep that in mind when evaluating with calculators

Limit Laws (Finite Right-Hand Side Results)

• If $\lim<em>{x\to a} f(x) = L$ and $\lim</em>{x\to a} g(x) = M$ with L, M finite, then:
  • Sum rule: $\lim_{x\to a} [f(x) + g(x)] = L + M$
  • Difference rule: $\lim_{x\to a} [f(x) - g(x)] = L - M$
  • Constant multiple rule: for any constant c,
    $\lim_{x\to a} [c \cdot f(x)] = c \cdot L$
  • Product rule: $\lim_{x\to a} [f(x) \cdot g(x)] = L \cdot M$
  • Quotient rule (provided denominator limit ≠ 0):
    $\lim_{x\to a} \frac{f(x)}{g(x)} = \frac{L}{M}$
  • Power rule (when defined):
    $\lim_{x\to a} [f(x)]^{\frac{m}{n}} = L^{\frac{m}{n}}$
• These rules are universal starting points; they extend to many standard limit computations

Squeeze Theorem (Sandwich Theorem)

• If $f(x) \le j(x) \le g(x)$ near a (except possibly at a) and
  $\lim<em>{x\to a} f(x) = \lim</em>{x\to a} g(x) = L$,
  then $\lim_{x\to a} j(x) = L$
• Commonly used when j(x) is trapped between two functions with the same limit

Delta-Epsilon (Definition of Limit) and a Simple Proof Strategy

• Formal definition:
  • For every \varepsilon > 0 there exists a \delta > 0 such that
    0 < |x - a| < \delta \quad\Rightarrow\quad |f(x) - L| < \varepsilon
• Interpretation:
  • “For every small tolerance in output, there is a small neighborhood around a where the function stays within that tolerance.”
• A typical proof outline (algebraic bounding):
  • Bound |f(x) - L| by a function of |x - a|, often by completing the square, factoring, or bounding auxiliary terms
  • Choose δ to enforce the bound, often as the minimum of several positive quantities to satisfy multiple inequalities
• Simple worked example (classic): prove lim_{x→0} f(x) = 0 for f(x) = x^2
  • |f(x) - 0| = |x^2| = |x|^2
  • If |x| < δ, then |f(x)| < δ^2
  • Given ε > 0, choose δ = min{1, √ε}; then if |x| < δ, |f(x)| < ε
• A more general technique mentioned in the transcript: bound |f(x) - L| by a multiple of |x - a| (or similar) and pick δ accordingly (e.g., taming via completing the square, bounding square roots, etc.). This is a common approach in more involved epsilon-delta proofs and connects to derivative intuition
• Note: The instructor indicates that delta-epsilon proofs are foundational for proving limits; they connect to subsequent calculus concepts like derivatives

Quick Practice and Notes on Limit Existence (Summary of Key Points)

• If left and right limits agree and are finite, the limit exists and equals that common value; the function value at the point may differ from the limit value
  • Example: left and right approach to 6 exist and equal 6, but f(6) may not be 6
• If the left and right limits do not agree, the limit does not exist (DNE)
• If both one-sided limits tend to the same infinity, the limit is infinite (not finite)
• If the one-sided limits are finite but different, the limit does not exist (jump discontinuity)
• If the function oscillates without settling near a point (e.g., f(x) = 6 cos(1/x) as x → 0), the limit does not exist
• When using limit laws, ensure the one-sided limits used in composite expressions exist and are finite where required
• In practice, both numerical evaluation (left/right approach) and analytical methods (factoring, series, known limit formulas) are used to determine limits

Real-World and Foundational Relevance

• Limits are the foundation for defining derivatives and integrals in calculus
• The “Welcome to calculus” moment refers to the derivative definition and the transition from limits to instantaneous rate of change
• Delta-Epsilon proofs, while theoretical, cement the rigorous meaning of limit concepts and underpin more advanced proofs in analysis

Final Quick References

• Two-sided limit existence criterion: $\lim<em>{x\to a} f(x) = L$ if $\lim</em>{x\to a^-} f(x) = \lim_{x\to a^+} f(x) = L\; (\text{finite})$
• Infinite limits: $\lim_{x\to a} f(x) = \infty\;\text{or}\; -\infty$ (not a finite value)
• Core limit laws: add, subtract, multiply by constant, multiply, divide (when denominator limit ≠ 0), and powers
• Squeeze Theorem: if f ≤ g ≤ h and lim f = lim h = L, then lim g = L
• Delta-Epsilon: formal definition and a simple illustrative example with a polynomial or basic function