Notes on Limits, Limit Laws, and Continuity (Transcript)

Summary of Key Ideas from the Transcript on Limits, Limit Laws, and Continuity

General mood: The instructor is guiding students through applying limit laws to various WebAssign problems, emphasizing when to use which rule, and when direct substitution is valid. There are also conceptual discussions of continuity, the squeeze theorem, and removable discontinuities.

Limit Laws: Core Rules Mentioned

Sum Law (for limits)
- If the limits
  
  ext{lim}{x\to a} f(x) and ext{lim}{x\to a} g(x)
  
  exist (i.e., are real numbers), then the limit of the sum is the sum of the limits:
  
  \lim{x\to a} [f(x) + g(x)] = \lim{x\to a} f(x) + \lim_{x\to a} g(x).
- This allows you to compute the limit of a sum by taking limits of the individual functions first.
Quotient Law (for limits)
- If the limits
  \lim{x\to a} f(x) = L and \lim{x\to a} g(x) = M exist, and M \neq 0, then
  \lim_{x\to a} \frac{f(x)}{g(x)} = \frac{L}{M}.
- Important caveat: you cannot conclude the limit exists or equal to L/M if the limit of the denominator is 0. If the denominator’s limit is 0 (or if the limit of either numerator or denominator doesn’t exist), the limit may not exist or require a different approach.
- The instructor notes that a WebAssign problem expecting you to apply the quotient law under a zero-denominator scenario would be flawed, so be cautious.
Product Law (implied in discussion)
- If \lim{x\to a} f(x) = L and \lim{x\to a} g(x) = M exist, then
  \lim_{x\to a} [f(x) \cdot g(x)] = L \cdot M.
- The instructor mentions you could apply the product rule similarly to other problems, provided the limits exist and you can use direct substitution or algebraic simplification first.
Exponent/Composition Considerations
- When a problem has a composite form like [f(x)]^n or a function inside another operation, you can often apply the limit to the inner function first, then apply the outer operation if the inner limit exists. The instructor mentions using an exponent rule to solve without worrying about the exponent for the moment, then decide whether to take the limit first or last.
- In some cases, you can decide the order (limit first vs. algebraic simplification) based on what exists or what is easiest to compute.
Direct Substitution vs. Algebraic Work
- If a function is continuous at a, then you can use direct substitution to evaluate the limit: \lim_{x\to a} f(x) = f(a).
- If direct substitution isn’t immediately possible, you can try algebraic simplification (cancellation, factoring) to reveal a removable discontinuity or to enable substitution.
- The instructor cautions that factoring to cancel common terms is a valid strategy but can add extra work; always first check whether direct substitution is already possible.

The Sandwich (Squeeze) Theorem

Statement (as presented):
- If three functions satisfy
  f(x) \le g(x) \le h(x)
  for all x near a, and
  \lim{x\to a} f(x) = \lim{x\to a} h(x) = l,
  then
  \lim_{x\to a} g(x) = l.
Intuition: If g is trapped between f and h, and both ends squeeze to the same limit, g must squeeze to that same limit as x approaches a.
Example scenario from the lecture:
- Suppose you want (\lim_{x\to 0} f(x)) given that
- 4 cos(2x) ≤ f(x) ≤ 3x^2 + 4 for x near 0.
- As x → 0, both 4 cos(2x) → 4 and 3x^2 + 4 → 4, so by the squeeze theorem, (\lim_{x\to 0} f(x) = 4).
Practical use: The teacher demonstrates drawing (or imagining) graphs and using direct substitution on the bounding functions to justify the limit of the middle function, especially when the middle function is difficult to pin down exactly but must lie between two well-behaved bounds.
Additional note on limits that are “in between”: If the bounds converge to a number l, you can deduce that the middle function’s limit must be l, but you may not be able to determine a precise value of g(x) itself without more information.

Continuity and Its Formal Definition

Informal idea: A function is continuous at a point a if you can draw the graph around a without lifting your pencil; there are no jumps or breaks.
Formal definition (as presented): A function f is continuous at a if
\lim_{x\to a} f(x) = f(a).
Three conditions (as highlighted in the lecture):
1) The limit as x approaches a exists and is a real number. (The limit must be a real number.)
2) The function value at a, i.e., f(a), is a real number (the function must be defined at a).
3) The limit equals the function value: \lim_{x\to a} f(x) = f(a).
The connection: If all three hold, the function is continuous at a; if any fail, continuity at a fails.
Visual intuition: A continuous function has outputs near a that are close to f(a), with no sudden jumps near a.

Discontinuities and Removable Discontinuities

Example scenario from the transcript:
- Consider a function defined such that at x = 1, f(1) = 3, but the surrounding outputs (as x → 1) approach 2.
- This is a removable discontinuity: the limit exists (equals 2), but the actual function value at the point is not equal to the limit.
- Graphically, there is a hole at (1, 2). If you redefine the value at x = 1 to be 2, the function becomes continuous at x = 1.
Key takeaway: Removable discontinuities occur when the limit exists but the function value at the point differs from that limit; they can be “fixed” by redefining the function value at the point.

Worked Concepts Discussed in the Transcript

Direct substitution as a first check
- If you can plug in the point a and get a real number, you often can conclude the limit (especially when the function is known to be continuous there or after simple simplification).
Graphical intuition
- The instructor suggests graphing or imagining the graphs to see where outputs tend as x approaches the target value, which can guide you to the correct limit or squeeze interval.
Recognizing when to avoid unnecessary algebra
- For instance, while factoring can sometimes help to cancel a problematic term, it may add extra steps; if a direct substitution is valid, it is often preferable.
Problem-solving workflow described in class
- Take time for private thinking to approach a problem (e.g., 30 seconds).
- Discuss approaches with neighbors to share rough ideas before formalizing the solution.
- In particular, for limit problems, determine whether to apply a limit law directly or to transform the expression first.

Practical Takeaways and Rules of Thumb

Always check the existence of the individual limits before applying a limit law that requires them.
For quotients, ensure the denominator limit is nonzero; otherwise, the limit may not exist or require a different approach.
When a function is continuous at a, you can use direct substitution: \lim_{x\to a} f(x) = f(a).
The squeeze theorem is a powerful tool when you can bound a function between two others whose limits are known and equal.
Removable discontinuities are common and can be fixed by redefining the function’s value at the point of discontinuity to equal the limit.
When solving limit problems, consider both algebraic manipulation (factoring, canceling) and substitution, and choose the simplest valid path first.

Quick Reference Formulas (LaTeX)

Sum law for limits:
\lim{x\to a} [f(x) + g(x)] = \lim{x\to a} f(x) + \lim_{x\to a} g(x)
Quotient law for limits (denominator limit nonzero):
\lim{x\to a} \frac{f(x)}{g(x)} = \frac{\lim{x\to a} f(x)}{\lim{x\to a} g(x)}\;\text{provided}\;\lim{x\to a} g(x) \neq 0.
Product law for limits:
\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).
Squeeze theorem:
If f(x) \le g(x) \le h(x) near a and \lim{x\to a} f(x) = \lim{x\to a} h(x) = L, then \lim_{x\to a} g(x) = L.
Continuity at a:
\lim_{x\to a} f(x) = f(a).
Removable discontinuity (concept): If \lim_{x\to a} f(x) = L exists but f(a) \neq L, the discontinuity at a is removable.

Examples You Might See on WebAssign (From the Transcript)

Example 1: Evaluate \lim{x\to 2} [f(x) + g(x)] using the Sum Law, given that both \lim{x\to 2} f(x) and \lim_{x\to 2} g(x) exist.
Example 2: A quotient limit where the denominator’s limit is nonzero, e.g., if \lim{x\to a} f(x) = L and \lim{x\to a} g(x) = M \neq 0, then use the Quotient Law to get \lim_{x\to a} \frac{f(x)}{g(x)} = \frac{L}{M}.
Example 3 (Squeeze): Given f(x) ≤ g(x) ≤ h(x) with \lim{x\to a} f(x) = \lim{x\to a} h(x) = L, conclude \lim_{x\to a} g(x) = L.
Example 4 (Continuity): If you can show that the limit exists and equals the function value, then the function is continuous at that point; if not, identify whether a removable discontinuity is present and how to fix it.

Note on Context

The transcript emphasizes a classroom approach: private think time, peer discussion, and progressively applying limit laws to different problem types. It also contrasts direct substitution with using limit laws, highlighting practical problem-solving strategies for limit questions on WebAssign and similar platforms.