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Flashcards · AP® Calculus BC · Unit 1

Unit 1: Limits and Continuity flashcards

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1.1 Introducing Calculus: Can Change Occur at an Instant?

What is the average rate of change of f over [a, b], and what are its units?
(f(b) − f(a))/(b − a): the change in output divided by the change in input. Its units are output units per input unit, and on a graph it is the slope of the secant line through (a, f(a)) and (b, f(b)).
Why can a rate of change at a single instant not be found with one division?
Over an interval of length zero the change in input is 0, so the average-rate formula gives (f(a) − f(a))/0 = 0/0, which is undefined (not 0).
How is the instantaneous rate of change at x = a obtained from average rates?
Use intervals that contain a and make them shorter. If the average rates from both sides approach one value, that value is the rate at the instant: lim (h → 0) (f(a + h) − f(a))/h. Graphically, secant slopes approach the tangent slope.
For f(x) = x², the average rate over [5, 5 + h] simplifies to 10 + h (h ≠ 0). What is the rate of change at x = 5?
10, because 10 + h approaches 10 as h approaches 0. You never put h = 0 into the quotient itself.
What are the common mistakes when finding a rate at an instant?
Putting h = 0 into the difference quotient (0/0); dividing an output by an input such as s(3)/3; using a long interval when a shorter one is available; averaging outputs instead of dividing changes; dropping units or the sign.
How do you estimate a rate at an instant from a table?
Use the shortest interval in the table that contains the instant, ideally with data on both sides, and compute (change in output)/(change in input). A table only gives an estimate.

1.2 Defining Limits and Using Limit Notation

What does lim (x → c) f(x) = R mean?
f(x) can be made as close to R as you like by taking x close enough to c, but not equal to c. Inputs on both sides of c count; x = c itself never counts. The limit is one real number.
Does the value f(c) affect lim (x → c) f(x)?
No. The limit ignores f(c). f(c) can equal the limit, differ from it, or be undefined. A hole does not stop a limit existing.
p(4) = 0 and lim (x → 4) p(x) = 9. Is this possible?
Yes. The graph has a hole at (4, 9) and a separate point at (4, 0). The limit describes where the outputs head, not the output at 4.
Does a table showing f(2.999) = 6.0004 and f(3.001) = 5.9996 prove lim (x → 3) f(x) = 6?
No. It suggests 6, but a table shows only a few inputs. A table supports an estimate, not a proof.
What notation mistakes are common with limits?
Writing lim f(x) = R with no x → c under lim; dropping lim so the statement becomes f(x) = R; swapping c (an input) and R (an output); using f(c) as the limit without a reason.
W(d) is water depth in cm, d days into a drought. How do you interpret lim (d → 20) W(d) = 15?
As the time gets closer to day 20, the depth gets as close as we like to 15 cm. Give the context and units.

1.3 Estimating Limit Values from Graphs

How do you read a limit from a graph?
Cover up x = a and follow the curve towards a from the left, then from the right. Each side gives a height (a y-value). Open circles, filled dots and gaps at a itself do not change the limit.
A graph has an open circle at (4, 2) and a filled dot at (4, 7). What is lim (x → 4) f(x)?
2, the height of the hole. The dot gives f(4) = 7, which does not affect the limit.
What are the three graphical reasons a limit fails to exist?
The left and right limits differ (jump); the function is unbounded (vertical asymptote); or it oscillates faster and faster.
When may you write lim (x → a) f(x) = ∞?
When both sides grow without bound upwards, for example lim (x → 1) 1/(x − 1)⁴ = ∞. It explains why the limit does not exist; ∞ is not a value.
What does x → a⁻ mean?
x approaches a from the left (values less than a). It does not mean x is negative.
What can you read at an endpoint of a function's domain?
Only one one-sided limit, the one from inside the domain.

1.4 Estimating Limit Values from Tables

How do you estimate a limit from a table of values?
Read the rows with x below c moving towards c (left-hand limit) and the rows with x above c moving towards c (right-hand limit). If both approach the same L, estimate the limit as L. Ignore f(c) even if shown.
Left values approach 3 and right values approach 1. What is the two-sided limit?
It does not exist, because the one-sided limits differ. It is not 2: never average two different one-sided values.
To how many decimal places should you give a limit estimate from a table?
To the places on which the closest values from both sides agree. For 3.1372 (left) and 3.1461 (right): 1 decimal place, 3.1.
Which x values make a good table for estimating lim (x → c) f(x)?
c ± 0.1, c ± 0.01, c ± 0.001, then c ± 0.0001 if more precision is needed. Work in radians for trig.
Why can a table mislead you about a limit?
It assumes the pattern continues between sampled points. For example sin(π/x) at x = 0.1, 0.01, 0.001 gives 0 each time, yet lim (x → 0) sin(π/x) does not exist.
What does a table with values 100, 10 000, 1 000 000 near c tell you?
The function is unbounded: there is no finite limit. You may write = ∞ (or −∞) only when both sides share one sign.

1.5 Determining Limits Using Algebraic Properties of Limits

What are the limit properties for sums, products and quotients?
If lim f = L and lim g = M both exist: sum/difference L ± M; constant multiple kL; product LM; quotient L/M only if M ≠ 0; power Lⁿ; root ⁿ√L (L > 0 for an even root).
How do you find the limit of a polynomial or rational function at x = a?
Substitute: lim (x → a) p(x) = p(a) for a polynomial, and for a rational function substitute as long as the denominator is not 0 at a.
lim (x → 1) f(x) = 2 and lim (x → 1) g(x) = −1. Find lim (x → 1) [(f(x))²g(x) − 3g(x)].
−1: 4 × (−1) − 3 × (−1) = −4 + 3 = −1.
When does a two-sided limit exist?
Only when both one-sided limits exist and are equal. The value f(a) never decides it.
If lim f does not exist, can lim (f + g) still exist?
Yes. The properties are silent when a separate limit does not exist; jumps can cancel. Work out each one-sided limit of the whole expression instead.
Find lim (x → −2) (x³ + x)/(x − 1).
10/3. The top tends to −8 − 2 = −10 and the bottom to −3, which is nonzero, so substitute.

1.6 Determining Limits Using Algebraic Manipulation

What does 0/0 from direct substitution tell you about a limit?
Nothing yet: 0/0 is an indeterminate form. Rewrite the expression (factor, conjugate, combine fractions, trig identity), then substitute again.
What does nonzero/0 from direct substitution tell you?
It is not indeterminate: the expression is unbounded near a, so there is no finite limit. Check the sign on each side. No algebra can rescue a finite limit.
Why is cancelling a factor (x − a) allowed when finding a limit at a?
Cancelling is valid only for x ≠ a, and a limit never uses the value at a, so the simplified expression has the same limit. Graphically the cancelled factor is a hole.
How do you handle a square-root difference that gives 0/0?
Multiply top and bottom by the conjugate, using (p − q)(p + q) = p² − q². Expand only the side with the root; leave the other side factored so the common factor can cancel.
Find lim (h → 0) ((3 + h)² − 9)/h.
6. The top is 6h + h², so the expression is 6 + h for h ≠ 0, which tends to 6.
Which factorisations are most useful for 0/0 limits?
x² − a² = (x − a)(x + a) and x³ − a³ = (x − a)(x² + ax + a²). Watch signs: 3 − x = −(x − 3).
What mistakes are common with algebraic limits?
Writing "0/0 so the limit is 0"; cancelling terms rather than whole factors; dropping lim before substituting; expanding the denominator after a conjugate; sign slips with a − x.

1.7 Selecting Procedures for Determining Limits

What is the first step for any limit given by a formula?
Substitute. A number with a nonzero bottom is the limit; 0 over a nonzero number gives 0; nonzero/0 means no finite limit (check signs); 0/0 means rewrite.
Which tool matches which 0/0 form?
Polynomial over polynomial: factor and cancel. Square root and a number: conjugate. Fraction inside a fraction: common denominator. Trig: an identity. Bounded oscillating factor such as cos(1/x): squeeze theorem.
How do you find a limit involving |x − a| or a piecewise rule split at a?
Find each one-sided limit with the rule that applies on that side. The limit exists only if they are equal. Never cancel |x − a| as if it were (x − a).
Find lim (x → −1) (x + 1)/|x + 1|.
It does not exist: the expression is 1 from the right and −1 from the left.
Describe (x² + 3)/(x − 1) as x → 1.
Substitution gives 4/0, so there is no finite limit: +∞ from the right and −∞ from the left, so the two-sided limit does not exist.
How do you find a composite limit lim f(g(x))?
Find the limit of the inside, L = lim g(x), then apply the outer function if it is continuous at L. Do not give up because the inside is undefined at a.

1.8 Determining Limits Using the Squeeze Theorem

What does the squeeze theorem state?
If g(x) ≤ f(x) ≤ h(x) for all x in an open interval around a (except possibly at a), and lim g = lim h = L, then lim (x → a) f(x) = L.
When should you use the squeeze theorem?
When the limit laws fail, usually because one factor such as sin(1/x) or cos(1/x) has no limit but is bounded between −1 and 1.
How do you find lim (x → 0) x² cos(7/x)?
−1 ≤ cos(7/x) ≤ 1, and x² ≥ 0, so −x² ≤ x² cos(7/x) ≤ x². Both bounds tend to 0, so by the squeeze theorem the limit is 0.
What are the two key trig limits from the squeeze theorem?
lim (x → 0) (sin x)/x = 1 and lim (x → 0) (1 − cos x)/x = 0. Both need radians.
Find lim (x → 0) (sin 6x)/(2x).
3. Write it as 3 × (sin 6x)/(6x): the angle and denominator must match, and (sin 6x)/(6x) → 1.
Why is −x ≤ x cos u ≤ x a wrong bound near 0?
It fails for x negative. Use |x cos u| ≤ |x|, so −|x| ≤ x cos u ≤ |x|, which holds on both sides of 0.
If the two outer bounds have different limits, what does the squeeze theorem tell you?
Nothing. It does not prove the limit fails to exist; there is simply no conclusion. The theorem also never gives the value f(a).

1.9 Connecting Multiple Representations of Limits

How do you read a limit from a graph, a table, a piecewise formula and words?
Graph: cover x = c and follow the curve from each side. Table: follow the output trend from each side. Piecewise: use the piece that applies on each side. Words: "approaches from the right" means lim (x → c⁺).
f(x) = 4 − x for x below 1, f(x) = x² + 2 for x above 1, f(1) = 0. Find lim (x → 1) f(x).
3. Left: 4 − 1 = 3. Right: 1 + 2 = 3. Both sides agree, and f(1) = 0 is irrelevant.
How do you find lim (x → c) f(g(x)) when f jumps at L = lim g(x)?
Find which side g(x) approaches L from, then use the matching one-sided limit of f at L, not f(L).
lim (u → 0⁻) f(u) = 5, lim (u → 0⁺) f(u) = −2, and g(x) → 0 from above as x → 4. Find lim (x → 4) f(g(x)).
−2. The input to f arrives from the right, so use the right-hand limit of f at 0.
Can (x + 1)·f(x) have a limit at c even if f does not?
Yes. Do not say "no limit" too early: the product property is silent when one limit fails, so split into one-sided limits of the whole expression.
How do you write "as t gets close to 6 from the right, D(t) gets close to 0.5" in notation?
lim (t → 6⁺) D(t) = 0.5.

1.10 Exploring Types of Discontinuities

What are the three types of discontinuity in this course?
Removable (left and right limits finite and equal, but f(c) undefined or different: a hole); jump (finite one-sided limits that differ); and vertical asymptote (f → ±∞ on at least one side).
In a rational function, how do you tell a hole from a vertical asymptote?
Factor. A factor (x − c) that cancels gives a hole; a factor that remains in the denominator gives a vertical asymptote. A zero denominator only shows where to look.
Classify the discontinuity of (x² − 25)/(x − 5) at x = 5.
Removable. It equals x + 5 for x ≠ 5, so the limit is 10, but the function is undefined at 5.
Classify the discontinuities of (x + 1)/(x² − 1).
It equals 1/(x − 1) for x ≠ −1. Removable at x = −1 (limit −1/2) and a vertical asymptote at x = 1.
lim (x → 2⁻) f(x) = 2, lim (x → 2⁺) f(x) = 6 and f(2) = 6. What type of discontinuity is this?
Jump: the one-sided limits are finite but differ. f(2) matching one side does not make it removable.
What decides the type of a discontinuity?
The one-sided limits, not just whether f(c) is defined. Check both sides before naming the type.

1.11 Defining Continuity at a Point

What three conditions make f continuous at x = c?
(1) f(c) exists; (2) lim (x → c) f(x) exists; (3) lim (x → c) f(x) = f(c). Check them in order and stop at the first failure.
How do you check continuity at the join of a piecewise function?
Find f(c) from the piece whose inequality includes c (≤ or ≥). Find both one-sided limits, each with its own piece. Continuous only if both limits equal f(c).
f(x) = 2x + 1 for x below 3 and f(x) = x² − 2 for x ≥ 3. Is f continuous at 3?
Yes. f(3) = 9 − 2 = 7; the left limit is 2(3) + 1 = 7 and the right limit is 7, so all three conditions hold.
f(c) = 2, lim (x → c⁻) f(x) = 2 and lim (x → c⁺) f(x) = 5. Which continuity condition fails?
Condition 2: the two-sided limit does not exist because the one-sided limits differ.
f(x) = (x² − 25)/(x − 5) for x ≠ 5 and f(5) = 10. Is f continuous at 5?
Yes. f(5) = 10, the limit is 5 + 5 = 10, and they are equal. A 0/0 formula does not by itself mean discontinuous.
What makes a continuity justification complete?
Quote values: f(c) = …, lim (x → c) f(x) = …, then state they are equal. "No break in the graph" with no numbers is not enough.

1.12 Confirming Continuity over an Interval

What does it mean for f to be continuous on a closed interval [a, b]?
f is continuous at every point of (a, b), plus lim (x → a⁺) f(x) = f(a) and lim (x → b⁻) f(x) = f(b). At an endpoint only the one-sided limit from inside is needed.
Which standard functions are continuous on their domains?
Polynomial, rational, power, exponential, logarithmic and trig functions. Sums, differences, products, quotients (nonzero denominator) and composites of continuous functions are also continuous.
Where are ln x, √x and tan x continuous?
ln x on (0, ∞); √x on [0, ∞); tan x and sec x everywhere except odd multiples of π/2; cot x and csc x everywhere except multiples of π.
Where is (x + 1)/(x² − 9) continuous?
On (−∞, −3), (−3, 3) and (3, ∞): everywhere except where x² − 9 = 0.
Is 1/x continuous on [−1, 1]?
No. "Continuous on its domain" is not the same as continuous on every interval: [−1, 1] contains 0, where 1/x is undefined.
Where is ln(x − 3) continuous?
On (3, ∞), because a logarithm needs x − 3 > 0.

1.13 Removing Discontinuities

How do you remove a removable discontinuity at x = c?
Find L = lim (x → c) f(x) (by algebra if 0/0), then define or redefine f(c) = L. You may change only the value at c, never the values near c.
f(x) = (x² − 49)/(x − 7) for x ≠ 7. What value of f(7) makes f continuous?
14, because the expression equals x + 7 for x ≠ 7, which tends to 14.
Can a jump discontinuity be removed by redefining f(c)?
No. If the one-sided limits differ (or are infinite) the two-sided limit does not exist, so no single value of f(c) works.
How do you choose a parameter to make a piecewise function continuous at a join c?
Substitute c into both expressions and set them equal (left value = right value). One parameter and one join give one equation; two parameters and two joins give simultaneous equations.
f(x) = kx − 1 for x below 2 and f(x) = x² + k for x ≥ 2. Find k for continuity.
2k − 1 = 4 + k, so k = 5. Both pieces then give 9 at x = 2.
What mistakes are common when removing discontinuities?
Saying "removable" without giving f(c); trying to fix a jump by picking one side; setting a piece equal to 0 instead of the other piece; substituting the wrong x; losing a solution of a quadratic in k.

1.14 Connecting Infinite Limits and Vertical Asymptotes

What is an infinite limit, and does it exist?
Values grow without bound near x = a: "= ∞" if positive, "= −∞" if negative. ∞ is not a number, so the limit does not exist; the symbol says how it fails.
When is x = a a vertical asymptote of f?
When at least one one-sided limit at a is ∞ or −∞. Justify in words, e.g. "Because lim (x → a⁺) f(x) = ∞, x = a is a vertical asymptote."
How does the power of (x − a) in a denominator affect the one-sided limits?
An odd power makes the sign flip across a, so the one-sided limits are opposite. An even power keeps the same sign on both sides, giving a two-sided ±∞.
Find the one-sided limits of (x + 4)/(x − 2) at x = 2.
Left: −∞; right: +∞. The top tends to 6 and x − 2 changes sign at 2.
Where does (x² − 4)/(x² + x − 6) have a hole and where a vertical asymptote?
It simplifies to (x + 2)/(x + 3) for x ≠ 2. Hole at (2, 4/5); vertical asymptote at x = −3.
What are standard infinite limits worth remembering?
lim (x → 0⁺) ln x = −∞; lim (x → π/2⁻) tan x = ∞ and lim (x → π/2⁺) tan x = −∞; lim (x → 0) 1/x² = ∞.
Is every zero of a denominator a vertical asymptote?
No. Cancel common factors first: some zeros are holes. Being undefined at a is not enough; you need an infinite limit.

1.15 Connecting Limits at Infinity and Horizontal Asymptotes

When is y = L a horizontal asymptote?
When f(x) → L as x → ∞ or as x → −∞. There are at most two, one per end, and the graph may cross a horizontal asymptote.
How do you find the limit at infinity of a rational function?
Compare degrees. Top lower: → 0 (asymptote y = 0). Equal: ratio of leading coefficients. Top higher: no finite limit and no horizontal asymptote.
Find lim (x → ∞) (3x − 7x³)/(2x³ + x).
−7/2: equal degrees, so take the ratio of the leading coefficients −7 and 2.
Find lim (x → −∞) √(x² + 5)/(2x).
−1/2. The root is positive and 2x is negative. For x → −∞, √(x²) = |x| = −x, so divide inside by −x.
What is the growth order of ln x, powers and exponentials for large x?
ln x grows slower than xᵖ (p > 0), which grows slower than bˣ (b > 1). So lim (x → ∞) (ln x)/x = 0.
How do you compare the growth of f and g as x → ∞?
Find lim (x → ∞) f(x)/g(x): ∞ means f dominates, 0 means g dominates, and a positive finite number means they grow at the same rate.
What are common mistakes with limits at infinity?
Substituting ∞ or treating ∞/∞ as 1; writing √(x²) = x when x → −∞; using constant terms instead of leading terms; calling y = ∞ an asymptote; assuming both ends match.

1.16 Working with the Intermediate Value Theorem (IVT)

What does the Intermediate Value Theorem state?
If f is continuous on the closed interval [a, b] and d is between f(a) and f(b), then there is at least one c between a and b with f(c) = d.
What does the Intermediate Value Theorem not tell you?
Where c is, or how many values there are: it guarantees at least one. If a condition fails, the value is "not guaranteed", not "impossible".
How do you show q(x) = x³ + 2x − 5 has a zero in (1, 2)?
q is a polynomial, so continuous on [1, 2]. q(1) = −2 and q(2) = 7, so q(1) is below 0 and q(2) above 0. By the Intermediate Value Theorem, q(c) = 0 for some c in (1, 2).
How do you use the Intermediate Value Theorem to show g(x) = h(x) has a solution?
Apply the theorem to g(x) − h(x) with target 0, showing it is continuous and changes sign on the interval.
f(x) = 1/x has f(−1) = −1 and f(1) = 1, but is never 0. Does this contradict the theorem?
No. f is not continuous at x = 0, so the theorem does not apply on [−1, 1].
What must a full Intermediate Value Theorem justification include?
A continuity statement with a reason (e.g. polynomial), the values f(a) and f(b), the inequality showing d is between them, and the conclusion naming the theorem.

Every card in this unit is shared with the AP® Calculus AB flashcards, because the two courses share these topics.

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