Limits and Continuity – Study Notes

What is a Limit (Page 1)

• Notation and idea
  • The limit of a function f(x) as x approaches a is written as
    $\lim_{x \to a} f(x) = L$
  • Intuition: as x gets arbitrarily close to a from both sides, f(x) gets arbitrarily close to L.
  • One-sided limits:
  • Right-hand limit: \lim_{x \to a^+} f(x) = L\n
  • Left-hand limit: $\lim_{x \to a^-} f(x) = L$
  • Existence of the two one-sided limits
  • If both one-sided limits exist and are equal to the same L, then the (two-sided) limit exists and equals L.
  • If they differ, the limit does not exist (DNE).
• Examples referenced in the transcript (illustrative patterns)
  • Example pattern: if as x approaches 4 from both sides, f(x) approaches 2, then
    $\lim_{x \to 4} f(x) = 2.$
  • Some worked examples shown include evaluating limits of simple expressions by substitution when the function behaves nicely around the point of interest.
• Important general ideas
  • The limit describes the value that f(x) approaches, not necessarily the value f takes at x = a.
  • If f is defined at a but the limit as x → a is not equal to f(a), f can be discontinuous at a (see Page 4).

The limit laws (Page 2)

• Suppose that $\lim<em>{x \to a} f(x) = F$ and $\lim</em>{x \to a} g(x) = G$ exist.
• Fundamental algebraic limit rules:
  1. Sum law:
     $\lim<em>{x \to a} [f(x) + g(x)] = \lim</em>{x \to a} f(x) + \lim_{x \to a} g(x) = F + G.$
  2. Difference law:
     $\lim<em>{x \to a} [f(x) - g(x)] = \lim</em>{x \to a} f(x) - \lim_{x \to a} g(x) = F - G.$
  3. Product law:
     $\lim<em>{x \to a} [f(x) \cdot g(x)] = (\lim</em>{x \to a} f(x)) \cdot (\lim_{x \to a} g(x)) = F \cdot G.$
  4. Quotient law:
     $\lim<em>{x \to a} \frac{f(x)}{g(x)} = \frac{\lim</em>{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{F}{G},$
     \quad provided (G \neq 0) (i.e., the denominator limit is not zero).
  5. Continuity/composition principle (informal form):
  • If f is continuous at L and (\lim{x \to a} g(x) = L), then $\lim</em>{x \to a} f(g(x)) = f(\lim_{x \to a} g(x)) = f(L).$
  • Corollary: If both f and g are continuous at a and g(a) is in the domain of f, then the composition f(g(x)) is continuous at a.
• Worked example (from transcript): polynomial limit via direct substitution
  • If (\lim{x \to 3} (3x^2 - 4x + 4)) is computed, use the limit laws or direct substitution (polynomials are continuous everywhere): $\lim</em>{x \to 3} (3x^2 - 4x + 4) = 3(3)^2 - 4(3) + 4 = 27 - 12 + 4 = 19.$
  • The transcript indicates plotting out terms like (3x^2), (-4x), and (+4) and pulling out constants, leading to the same result.
• Notes about continuity and limits in examples
  • If the limit exists (one of the laws above applies) you can substitute the limit values for x when appropriate.
  • The concept of composition is especially useful for evaluating limits of more complex functions built from simpler, known-continuous pieces.

Computing limits using algebraic methods (Page 3)

• Direct substitution and continuity (basic rule)
  • If a function f is continuous at a, then
    $\lim_{x \to a} f(x) = f(a).$
  • This is often referred to as the direct substitution rule when continuity holds at a.
• Theorem: Continuity of standard functions on their domains
  • Constant functions: (f(x) = C). Continuous everywhere on their domain.
  • Identity function: (f(x) = x). Continuous everywhere.
  • Power functions: (f(x) = x^n) (n typically a real number; for integer n this is standard). Continuous on their domains.
  • Exponential functions: (f(x) = a^x) with (a>0). Continuous on (\mathbb{R}).
  • Logarithmic functions: (f(x) = \log_a x) with (a>0, a \neq 1). Continuous on ((0, \infty)).
  • Trigonometric functions: (\sin x), (\cos x) are continuous on (\mathbb{R}).
• Example: Direct substitution for a continuous function
  • If we have a limit like
    $\lim_{x \to a} f(x) = f(a)$
    and f is continuous at a, we substitute directly to obtain the limit.
• Approach from left and right
  • In some cases you compute the left-hand limit and right-hand limit separately and then compare:
    $\lim<em>{x \to a^-} f(x) = L</em>1, \quad \lim<em>{x \to a^+} f(x) = L</em>2.$
  • If (L1 = L2 = L), then the two-sided limit exists and equals L.
  • If (L1 \neq L2), the two-sided limit does not exist (DNE).
• Special limit forms and examples mentioned in transcript
  • Example: (\lim_{x \to 0} \dfrac{\sin x}{x} = 1) (standard limit derived from continuity properties of sin and x near zero).
  • Infinite discontinuity example: as x approaches a point where the function grows without bound, the limit is (\infty) or (-\infty) or DNE depending on direction.
  • Polynomial and elementary functions are typically continuous at all points in their natural domains, so direct substitution suffices for limits at those points.

Definition of continuity and types of discontinuities (Page 4)

• Definition (formal): A function f is continuous at a point a if
  • f(a) is defined,
  • the limit (\lim_{x \to a} f(x)) exists, and
  • these two values are equal:
    $\lim_{x \to a} f(x) = f(a).$
• A function is continuous on an interval if it is continuous at every point in that interval.
• One-sided and two-sided continuity
  • Left-continuity: lim from the left equals f(a) when approaching a from the left.
  • Right-continuity: lim from the right equals f(a) when approaching a from the right.
• Types of discontinuities (conceptual)
  • Removable discontinuity at a: The limit lim_{x→a} f(x) exists (finite) but f(a) is either not defined or not equal to that limit.
  • Jump discontinuity at a: The left and right limits exist but are not equal to each other.
  • Infinite discontinuity at a: At least one of the one-sided limits diverges to ±∞.
• Continuity on intervals with holes
  • A function can be continuous on intervals that are unioned, such as [0,4] ∪ (4,6) ∪ (6,10], where the points of discontinuity (e.g., x=4, x=6) are excluded from the interval of continuity.
• Examples and notations from transcript (illustrative points)
  • At some points in example graphs, certain x-values like x = 1, x = 4, x = 6, and x = 10 are discussed in terms of continuity or discontinuity.
  • A typical assessment would check left/right limits, the value of the function at the point, and graph behavior to classify the type of discontinuity.

Practice problems and problem-solving approach (Page 5)

• General strategy for evaluating limits
  • Identify if the function is continuous at the target point. If yes, use direct substitution:
    $\lim_{x \to a} f(x) = f(a).$
  • If not continuous, attempt to compute one-sided limits:
    $\lim<em>{x \to a^-} f(x), \quad \lim</em>{x \to a^+} f(x).$
  • If both one-sided limits exist and are equal, that common value is the limit; otherwise the limit DNE.
  • When the expression is a quotient, ensure the denominator limit is nonzero; otherwise the limit may be infinite or DNE.
  • For piecewise functions, evaluate the limit from each piece and ensure consistency at the target point.
• Types of limits you may encounter in problems
  • Constant and linear functions: direct substitution usually works.
  • Polynomials and rational functions: direct substitution unless a denominator zero occurs or there is a removable discontinuity.
  • Trigonometric and exponential/logarithmic functions: use known continuity properties or standard limits.
  • One-sided limits and removable discontinuities: determine if a hole can be removed by redefining the function at the point.
• Worked example to consolidate the approach
  • Example 1: Direct substitution for a polynomial at a point where the function is continuous:
    $\lim_{x \to 3} (3x^2 - 4x + 4) = 19.$
  • Example 2: If (\lim{x \to a} g(x) = L) and g is approaching L from both sides, and f is continuous at L, then $\lim</em>{x \to a} f(g(x)) = f(L).$
• Quick reminders from the transcript
  • A limit exists only if both one-sided limits exist and agree.
  • A limit that tends to infinity is described with ±∞ depending on direction, and can indicate an infinite discontinuity.
  • Removable vs jump discontinuities are distinguished by whether the limit exists and whether f(a) equals that limit.
• Summary of how the content ties together
  • Limits provide a rigorous way to talk about the value a function approaches near a point.
  • Continuity is a stronger condition stating the function value at a point matches the limiting value as x approaches that point.
  • Many standard function classes (constants, polynomials, exponentials, logs, trig functions) are continuous on their natural domains, which makes limit calculation via direct substitution straightforward for those points.