Comprehensive Study Notes: Limits (Sections 1.1–1.10)

1.1 Can Change Occur at an Instant?

Central idea: instantaneous rate of change vs. average rate of change. The instantaneous rate is what a derivative gives at a specific point, while the average rate is the slope of a line between two points.
Tangent line intuition: the slope of the tangent at a given input value represents the instantaneous rate of change there.
General approach:
- If a function models views v(w) with w = weeks since the channel started, the tangent at w = 10 gives the instantaneous rate of change of views per week at week 10.
- If you cannot compute a derivative directly, approximate the tangent slope using nearby values: e.g., slope ≈ [f(a+h) − f(a)]/h for small h, or use a closer pair like [f(a) − f(a − h)]/h.
Example style problems (from transcript):
- Mr. Kelly’s channel views v(w): draw tangent at w = 10; estimate instantaneous rate of change there.
- For distance D(t): average speed over an interval is the average rate of change, computed as $\text{Average rate} = \frac{D(t2) - D(t1)}{t2 - t1}.$
- If the trip ends at t = 8 minutes, the average speed to the store is $\frac{D(8) - D(0)}{8 - 0}.$ (Use appropriate starting time and distance values from the graph.)
Specific numeric examples (as seen in the transcript):
- A rough instantaneous rate between two close times can be estimated by a secant-like quotient: e.g., between t = 2 and t = 2.001, or t = 9 and t = 9.001, etc.
- Example tangent slope shown: 150 m/min for a rough instantaneous rate between two close minutes, illustrating the idea that instantaneous rate is the limit of average rates as the interval shrinks.
Buffalo population example (for interpreting limits):
- If b(t) is the buffalo population (t in years since 1800), then
- $b(90)$ represents the buffalo population in the year 1890.
- An approximate instantaneous rate near t = 32 can be obtained by
  $\frac{b(32) - b(31.999)}{32 - 31.999}.$
Raspberry bush height example: h(t) is height in feet, t is weeks since planting.
- To estimate the instantaneous rate at t = 9, use a small step:
  $\text{Rate} \approx \frac{h(9) - h(8.999)}{9 - 8.999}.$
- A sample calculation in the transcript shows a rough rate like $0.083\ \text{ft/week}$ (from two closely spaced data values).
Amusement park entries E(t): E(t) is the number of people (in thousands) entering since 10:00 a.m.; t is hours since 10:00 a.m.
- The tangent line at a chosen time (e.g., t = 3) gives the instantaneous rate:
  $\text{Rate}_{\text{instant}} \approx \frac{E(3) - E(2.999)}{3 - 2.999}.$
Free throws problem (season model): If L(g) is the total number of made free throws over g games (0 ≤ g ≤ 82), then
- L(50) represents the total makes in 50 games.
- A rough average rate of change up to game 50 is
  $\frac{L(50) - L(0)}{50 - 0}.$
Electric bill example: k(m) is kWh used in month m (0 ≤ m ≤ 12).
- k(8) represents the kWh used in the 8th month; the average change from month 5 to month 8 is
  $\frac{k(8) - k(5)}{8 - 5}.$
Deaths example: d(t) is the number of deaths per year with t = years since 1950 (0 ≤ t ≤ 50).
- d(40) represents deaths in the year 1990 (if t = 40 corresponds to 1990).
- The average rate of change from 1960 to 1970 is
  $\frac{d(20) - d(10)}{20 - 10}.$
Dam release example: V(t) is cubic liters released since opening; t is seconds (0 ≤ t ≤ 3600).
- V(100) is the amount released after 100 seconds.
- The average rate of change from 0 to 100 seconds is
  $\frac{V(100) - V(0)}{100 - 0}.$
Takeaway: The instantaneous rate of change is the derivative at a point, accessible via tangent slopes or limit of average rates over shrinking intervals. When graphs or tables are available, approximate using nearby values and one-sided limits when needed.

1.2 Defining Limits

Core statement: A limit describes where f(x) is headed as x approaches a, not necessarily the actual function value at x = a.
Formal interpretation (conceptual):
- lim_{x\to a} f(x) = L means: as x gets arbitrarily close to a (but not equal to a), f(x) gets arbitrarily close to L.
Simple interpretations (from notes):
- lim_{x\to a} f(x) = L implies f(x) is near L when x is near a (but not necessarily equal to a).
- Limits can exist even if f(a) is undefined or f(a) ≠ L.
One-sided limits (context from graph problems):
- Left-hand limit: $\lim{x\to a^-} f(x) = L{-}$
- Right-hand limit: $\lim{x\to a^+} f(x) = L{+}$
- If $L{-} = L{+} = L$ , then the (two-sided) limit $\lim_{x\to a} f(x) = L$ exists; otherwise the limit does not exist (DNE).
Examples and evaluation patterns (from graph-based problems):
- If the left-hand limit equals -1 and the right-hand limit equals 2 as x approaches 3, then
  $\lim_{x\to 3} f(x) \quad \text{DNE}.$
- If both one-sided limits exist and equal the same value, the limit exists and equals that value.
Important conceptual notes:
- The limit focuses on the behavior of f(x) near a, not necessarily the value f(a).
- If a is an interior point with a continuous function, lim_{x\to a} f(x) = f(a).
Practice problem interpretations (from notes):
- Examples ask:
- What does lim_{x\to a} f(x) = L mean in plain terms?
- What is f(a) when the limit exists or does not exist?
Useful general guidelines:
- If the graph shows a hole at x = a, the limit often exists and equals the y-value of the graph approaching the hole, even if f(a) is not defined at a.
- If the graph has a jump around a, the left and right limits differ, so the limit does not exist.

1.3 Finding Limits from Graphs

One-sided limits from a graph: evaluate the y-value the graph approaches from the left and from the right as x approaches a.
If the two one-sided limits agree, the limit exists and equals that common value; if not, the limit does not exist.
Notation:
- Left limit: $\lim{x\to a^-} f(x) = L{-}$
- Right limit: $\lim{x\to a^+} f(x) = L{+}$
- Two-sided limit exists iff $L{-} = L{+} = L$ .
Example patterns from notes:
- The limit of f as x approaches 3 from the left is -1.
- The limit of f as x approaches 3 from the right is 2.
- If left and right limits are different, the overall limit does not exist.
Practice (from notes):
- For problems labeled 1–3, determine left-hand limit, right-hand limit, and the two-sided limit if possible; otherwise state DNE.
Graph-based reminders:
- A point can be on the graph (f(a) defined) or not; limits ignore the actual value at a, only the approach behavior.

1.4 Finding Limits from Tables

Rationale: When a graph is not available, a table of values around a is used to estimate the limit as x approaches a.
Strategy:
- Build a table of x-values increasingly close to a from both sides (e.g., a − Δ, a − Δ/2, a, a + Δ/2, a + Δ, …), and observe the corresponding f(x) values.
- If f(x) values approach a single number L from both sides, then lim_{x\to a} f(x) = L.
Calculator notes: Tables generated by calculators can provide quick approximations, but you should verify convergence and be mindful of rounding.
Example workflow (as in notes):
- Given a function, create a table around the target x-value and observe the trend of f(x) as x approaches that value.
- If the table shows values approaching, say, 21 from both sides, then lim_{x\to a} f(x) = 21.
Practice tasks (from notes) emphasize:
- Interpreting table entries to identify the limit value.
- Constructing your own table of values to approximate desired limits.

1.5 Algebraic Properties of Limits and Piecewise Functions

Core limit laws (provided in notes):
- Linear: $\lim{x\to a} [f(x) + g(x)] = \lim{x\to a} f(x) + \lim_{x\to a} g(x)$
- Scalar multiplication: $\lim{x\to a} [c \, f(x)] = c \cdot \lim{x\to a} f(x)$
- Difference: $\lim{x\to a} [f(x) - g(x)] = \lim{x\to a} f(x) - \lim_{x\to a} g(x)$
- Product: $\lim{x\to a} [f(x) g(x)] = (\lim{x\to a} f(x))(\lim_{x\to a} g(x))$
- Quotient (when the denominator limit is nonzero): $\lim{x\to a} \frac{f(x)}{g(x)} = \frac{\lim{x\to a} f(x)}{\lim_{x\to a} g(x)}$
Piecewise functions and limits:
- To compute the limit at a point where a piecewise definition changes, examine the left-hand and right-hand limits at that point.
- If both exist and are equal, the limit exists; otherwise it does not.
Function composition and limits (from notes):
- Example: If you know lim f(x) as x approaches a, and lim g(y) as y approaches f(a), you may consider lim f(g(x)) as x approaches a, when appropriate and defined.
- A graph example in notes showed lim f(f(x)) could exist even when f is not continuous at certain points.
Piecewise functions and limit behavior:
- The notes include several problems where the function is defined piecewise with different functional forms on different intervals; the limits may exist at boundary points depending on left- and right-hand values.
Takeaways:
- Use limit laws to simplify and evaluate limits when possible.
- Be mindful of domain issues where limits exist but the function is not defined at the point.

1.6 Algebraic Manipulation (Limits)

Direct substitution: If f is continuous at a, then lim_{x\to a} f(x) = f(a).
When substitution yields an indeterminate form, apply algebraic tricks:
- Factoring and canceling common factors.
- Rationalizing expressions with radicals (multiply numerator and denominator by the conjugate).
- Using conjugates to simplify expressions like (\sqrt{x+7} - 3)/(x - 5).
Standard small-angle and trig limits (used in the notes):
- $\lim_{x\to 0} \frac{\sin x}{x} = 1$
- $\lim_{x\to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$
- $\lim_{x\to 0} \frac{\tan x}{x} = 1$
Other technique examples from notes:
- Combine fractions to simplify or cancel, e.g., write over a common denominator and cancel common factors.
- Use known limits and algebraic rearrangements to resolve limits that initially appear as 0/0 or ∞/∞ forms.
Practice themes from notes:
- Limit problems involving rational expressions, products, quotients, and compositions.
- Use standard trig and algebraic limits to reduce to computable forms.

1.7 Selecting Procedures (Limits)

Core idea: choose an appropriate method based on the form of the limit expression.
Common procedures:
- Direct substitution when f is continuous at a.
- Algebraic simplification (factoring, expanding, combining fractions).
- Rationalizing with conjugates when radicals appear in a difference quotient.
- Handling complex fractions by simplifying step by step.
- Using conjugates to simplify expressions like (\sqrt{a} - \sqrt{b})/(a - b).
Examples (from notes):
- lim_{x\to 5} (\sqrt{x+7} - 3)/(x-5) can be solved by multiplying numerator and denominator by the conjugate (\sqrt{x+7} + 3).
- Complex fractions: simplify by multiplying numerator and denominator by a suitable expression to remove nested fractions.
Heuristics:
- If you see a radical difference in the numerator over a linear term, try a conjugate.
- If you see a ratio that gives 0/0, look for common factors to cancel or a substitution to simplify.
- When a product or quotient involves limits, check if you can split into known limit laws.

1.8 The Squeeze Theorem

Statement (Sandwich Theorem): If g(x) ≤ f(x) ≤ h(x) for all x near a (except possibly at a) and
- $\lim{x\to a} g(x) = \lim{x\to a} h(x) = L,$
 then
- $\lim_{x\to a} f(x) = L.$
Typical use cases (from notes):
- If |f(x)| ≤ |x| and lim{x\to 0} x = 0, then lim{x\to 0} f(x) = 0 by squeezing.
- For sin and cos bounded values, e.g., if -1 ≤ cos x ≤ 1, then lim_{x\to 0} x^2 cos x = 0 because -x^2 ≤ x^2 cos x ≤ x^2 and x^2 → 0.
Practice examples from notes:
- lim x cos x as x → 0 is 0 because cos x is bounded and x → 0.
- If g(x) ≤ f(x) ≤ h(x) and both g and h tend to the same limit, then f tends to that limit as x → a.
Important caveats:
- The Squeeze Theorem requires the bounding functions to converge to the same finite limit L.
- It cannot be used if the bounds do not share the same limit or if one bound diverges.

1.9 Multiple Representations (Limits)

Concept: Limits can be represented and analyzed via multiple representations: graphs, tables, analytical expressions, and symbolic manipulation.
Key ideas from notes:
- A piecewise function might have different limits from the left and right at a boundary; the limit exists only if these agree.
- If a function is defined in a way that creates a hole, the limit can exist independent of the function value at that point.
- The same limit can be approached using different representations (e.g., tabular data, algebraic manipulation, and graph inspection) and should yield consistent results.
Practice themes from notes:
- Evaluate limits using a combination of algebraic techniques, graph insights, and table-based approximations.
- Determine whether limits exist, and if so, compute their values; if not, justify DNE.
- For composite expressions, apply limit laws carefully and check for continuity where substitution would be valid.

1.10 Types of Discontinuities

Types of discontinuities:
- Removable discontinuity (hole): The limit exists, but the function is either undefined at that point or defined with a different value. Occurs when numerator and denominator share a common factor that can be canceled, leaving a finite limit at x = a after cancellation.
- Infinite discontinuity: The function grows without bound (±∞) as x approaches a.
- Jump discontinuity: The left-hand and right-hand limits exist but are not equal, causing a jump in the function value at a.
How to identify on graphs:
- Look for holes (circles without dots) indicating removable discontinuities.
- Look for vertical asymptotes where the function shoots to ±∞ (infinite discontinuity).
- Look for two different heights approaching from left and right indicating a jump.
Notes and practice from the transcript include multiple functions and graphs labeled to identify:
- Locations of discontinuities (x-values where the function is not continuous or limits fail to exist).
- Whether a discontinuity is removable, infinite, or a jump.
- Intervals of continuity for piecewise definitions.
Examples from notes (summary patterns):
- Some functions show a hole at x = -2, a hole at x = 0, or a jump at x = 1.
- Others display vertical asymptotes indicating infinite discontinuities at certain x-values.
Practical implications:
- Understanding discontinuities helps in choosing appropriate limit techniques and recognizing when limits equal function values (continuity) or not.

Quick reference: key limit concepts and formulas

Definition of a limit (informal):
- As x approaches a, f(x) approaches L if we can make f(x) arbitrarily close to L by taking x sufficiently close to a (but not equal to a).
Limit notation:
- Two-sided: $\lim_{x\to a} f(x) = L$
- One-sided: $\lim{x\to a^-} f(x) = L\, ,\quad \lim{x\to a^+} f(x) = L$
Limit laws (provided in notes):
- $\lim{x\to a} [f(x) + g(x)] = \lim{x\to a} f(x) + \lim_{x\to a} g(x)$
- $\lim{x\to a} [c f(x)] = c \cdot \lim{x\to a} f(x)$
- If both limits on the right exist and the denominator limit is nonzero: $\lim{x\to a} \frac{f(x)}{g(x)} = \frac{\lim{x\to a} f(x)}{\lim_{x\to a} g(x)}$
Fundamental small-angle limits (sample):
- $\lim_{x\to 0} \frac{\sin x}{x} = 1$
- $\lim_{x\to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$
- $\lim_{x\to 0} \frac{\tan x}{x} = 1$
Squeeze theorem (sandwich): if g(x) ≤ f(x) ≤ h(x) near a and lim g(x) = lim h(x) = L, then lim f(x) = L.
Important interpretation reminder:
- Limits do not necessarily equal the function value at the point; they describe behavior as x approaches the point.

Tips for exam prep from the transcript

Be comfortable with both graph-based and table-based limit thinking.
Practice distinguishing between the value of the function at a point and the limit as x approaches that point.
Master the standard limit tricks: factoring, conjugates, and known limits such as sin/x and (1 − cos x)/x^2.
When faced with a limit that yields 0/0 or ∞/∞, try algebraic simplification first before concluding DNE.
For piecewise definitions, check left- and right-hand limits at interval boundaries to decide if the overall limit exists.
Use the Squeeze Theorem whenever limits can be bounded tightly by functions with a known limit.

If you want, I can tailor these notes further into shorter flash-card style prompts or expand any section with step-by-step worked examples similar to the ones in the transcript.