AP Calculus AB · Flashcards
Unit 1 — Limits & Continuity
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Limit Laws✓ Known #1
When is it valid to evaluate a limit by direct substitution?
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Answer#1
- If the function is continuous at the point (the expression stays defined there).
- Polynomials are continuous everywhere; rational functions are continuous wherever the denominator \(\neq 0\).
Trap: Substituting into a rational expression where the denominator becomes \(0\).
Limit Laws✓ Known #2
If \(\lim_{x\to a} f(x)=L\) and \(\lim_{x\to a} g(x)=M\), what is \(\lim_{x\to a} [f(x)+g(x)]\)?
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Answer#2
\(\lim_{x\to a}[f(x)+g(x)]=L+M\).
Trap: Applying the laws when one of the limits does not exist.
Limit Laws✓ Known #3
What is \(\lim_{x\to a} [c\cdot f(x)]\) if \(c\) is a constant and \(\lim_{x\to a} f(x)=L\)?
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Answer#3
\(\lim_{x\to a} [c\,f(x)] = cL\).
Trap: Forgetting to apply the constant to the whole limit.
Limit Laws✓ Known #4
State the product and quotient limit laws (and when they apply).
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Answer#4
- \(\lim (fg) = (\lim f)(\lim g)\), if both limits exist.
- \(\lim \dfrac{f}{g}=\dfrac{\lim f}{\lim g}\), if both exist and \(\lim g\neq 0\).
Trap: Using the quotient law when \(\lim g=0\).
Limit Laws✓ Known #5
If \(\lim_{x\to a} f(x)=L\), what is \(\lim_{x\to a} [f(x)]^n\) for a positive integer \(n\)?
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Answer#5
\(\lim [f(x)]^n = L^n\).
Trap: Confusing \([f(x)]^n\) with \(f(nx)\).
Limit Laws✓ Known #6
If \(\lim_{x\to a} f(x)=L\) and \(f(x)\ge 0\) near \(a\), what is \(\lim_{x\to a} \sqrt{f(x)}\)?
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Answer#6
\(\lim \sqrt{f(x)}=\sqrt{L}\) (here \(L\ge 0\)).
Trap: Taking \(\sqrt{L}\) when \(L<0\) (not real).
Limit Laws✓ Known #7
What is \(\lim_{x\to a} |f(x)|\) if \(\lim_{x\to a} f(x)=L\)?
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Answer#7
\(\lim |f(x)| = |L|\).
Trap: Dropping the absolute value without considering the sign.
Limit Laws✓ Known #8
Why do we often rewrite expressions before taking a limit?
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Answer#8
- Indeterminate forms like \(0/0\) or \(\infty/\infty\) require simplification.
- Rewriting can reveal a removable factor or a simpler equivalent expression.
Trap: Treating indeterminate forms as actual values.
One-Sided Limits✓ Known #9
Define the left-hand and right-hand limits at \(x=a\).
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Answer#9
- Left: \(\lim_{x\to a^-} f(x)\) — \(x\) approaches \(a\) from values less than \(a\).
- Right: \(\lim_{x\to a^+} f(x)\) — \(x\) approaches \(a\) from values greater than \(a\).
Trap: Thinking \(a^-\) means “negative \(a\).”
One-Sided Limits✓ Known #10
When does the two-sided limit \(\lim_{x\to a} f(x)\) exist?
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Answer#10
- \(\lim_{x\to a^-} f(x)\) exists,
- \(\lim_{x\to a^+} f(x)\) exists,
- and they are equal.
Trap: “Both exist ⇒ the limit exists.” They must also be equal.
One-Sided Limits✓ Known #11
What does a jump discontinuity look like in limit language?
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Answer#11
Both one-sided limits exist (finite), but \(\lim_{x\to a^-} f(x)\neq \lim_{x\to a^+} f(x)\).
Trap: Writing a two-sided limit anyway.
One-Sided Limits✓ Known #12
Piecewise function at \(x=a\): what should you compute first?
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Answer#12
- \(\lim_{x\to a^-} f(x)\) using the left piece.
- \(\lim_{x\to a^+} f(x)\) using the right piece.
- Then compare them; check \(f(a)\) separately.
Trap: Plugging \(a\) into the wrong piece.
One-Sided Limits✓ Known #13
If \(\lim_{x\to a^-} f(x)=\infty\), what does that mean?
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Answer#13
As \(x\to a^-\), \(f(x)\) increases without bound: vertical-asymptote behavior from the left.
Trap: Treating \(\infty\) as a number.
One-Sided Limits✓ Known #14
Quick check: \(\lim_{x\to 0^-} \frac{|x|}{x}\) and \(\lim_{x\to 0^+} \frac{|x|}{x}\)?
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Answer#14
- \(x\to 0^-\): \(|x|=-x\Rightarrow \frac{|x|}{x}=-1\)
- \(x\to 0^+\): \(|x|=x\Rightarrow \frac{|x|}{x}=1\)
- So the two-sided limit does not exist.
Trap: Treating \(|x|\) as always equal to \(x\).
Infinite Limits✓ Known #15
What does \(\lim_{x\to a} f(x)=\infty\) indicate?
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Answer#15
Unbounded growth from both sides near \(x=a\) — the line \(x=a\) is a vertical asymptote. (Strictly, the limit does not exist as a number.)
Trap: Assuming two-sided behavior without checking both one-sided limits.
Infinite Limits✓ Known #16
If the left limit \(\to\infty\) and the right limit \(\to-\infty\), can we write a single two-sided limit?
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Answer#16
No. The one-sided behaviors differ, so the two-sided limit does not exist (not even as \(\pm\infty\)).
Trap: “Both are infinite, so the limit is infinite.”
Infinite Limits✓ Known #17
Compare \(\lim_{x\to 0}\frac{1}{x}\) and \(\lim_{x\to 0}\frac{1}{x^2}\).
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Answer#17
- \(\frac{1}{x}\): left \(\to -\infty\), right \(\to \infty\) ⇒ the two-sided limit DNE.
- \(\frac{1}{x^2}\): both sides \(\to \infty\).
Trap: Forgetting the sign change when the power is odd.
Infinite Limits✓ Known #18
How do you distinguish a hole from a vertical asymptote in a rational function?
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Answer#18
- Factor and cancel common factors first.
- A factor that cancels ⇒ removable discontinuity (hole).
- A zero of the denominator that remains after simplifying ⇒ vertical asymptote.
Trap: Declaring an asymptote before simplifying.
Infinite Limits✓ Known #19
If \(f(x)=\frac{1}{(x-a)^3}\), what happens as \(x\to a^-\) and \(x\to a^+\)?
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Answer#19
- \(x\to a^-\): \((x-a)^3<0\Rightarrow f(x)\to -\infty\)
- \(x\to a^+\): \((x-a)^3>0\Rightarrow f(x)\to \infty\)
Trap: Treating odd powers like even powers.
Infinite Limits✓ Known #20
If \(\lim_{x\to a^+} f(x)=\infty\), how do you say it in words (AP language)?
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Answer#20
“As \(x\) approaches \(a\) from the right, \(f(x)\) increases without bound.”
Trap: Writing \(\infty\) as a value instead of describing behavior.
Indeterminate Forms✓ Known #21
Substitution gives \(\frac{0}{0}\). What should you try first?
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Answer#21
- Factor and cancel.
- Combine into a single fraction (common denominator).
- Multiply by a conjugate (radicals).
Trap: Writing “\(0/0\)” as the answer.
Indeterminate Forms✓ Known #22
What does the form \(\frac{\infty}{\infty}\) mean?
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Answer#22
It is indeterminate: the limit could be \(0\), a nonzero number or \(\pm\infty\). You must rewrite or simplify.
Trap: “Infinity over infinity equals 1.”
Indeterminate Forms✓ Known #23
Is the form \(\frac{1}{0}\) indeterminate?
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Answer#23
No. A nonzero numerator over a denominator approaching 0 signals unbounded behavior (an infinite limit or vertical asymptote); check the one-sided signs.
Trap: Calling every zero denominator “\(0/0\).”
Indeterminate Forms✓ Known #24
If \(\lim f=0\) and \(\lim g=0\), what can you conclude about \(\lim \frac{f}{g}\)?
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Answer#24
Nothing yet — \(\frac{0}{0}\) is indeterminate. Rewrite or simplify first.
Trap: Concluding the limit is 1 because “both go to 0.”
Indeterminate Forms✓ Known #25
If a factor \((x-a)\) cancels, what does that usually mean about the original function at \(x=a\)?
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Answer#25
- The original function has a removable discontinuity (a hole) at \(x=a\).
- The limit can exist even though \(f(a)\) is undefined.
Trap: Calling it a vertical asymptote before canceling.
Indeterminate Forms✓ Known #26
What is the goal of rewriting an indeterminate form?
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Answer#26
- Find an expression equal to \(f(x)\) for \(x\neq a\) where substitution works.
- Or reveal one-sided unbounded behavior.
Trap: Doing algebra that changes the function.
Indeterminate Forms✓ Known #27
Is the form \(\frac{0}{5}\) indeterminate? What is the value?
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Answer#27
Not indeterminate. The limit is \(0\).
Trap: Calling a form indeterminate whenever you see a zero.
Indeterminate Forms✓ Known #28
Is the form \(\frac{\infty}{5}\) indeterminate? What is the behavior?
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Answer#28
Not indeterminate. The expression grows without bound (\(\to\infty\)).
Trap: Treating \(\infty\) as a number you can cancel.
Algebraic Techniques✓ Known #29
Technique: factoring. What is the typical trigger?
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Answer#29
Substitution gives \(0/0\) and you see polynomials: factor to cancel a common factor (often \(x-a\)).
Trap: Canceling terms that are added — only common factors cancel.
Algebraic Techniques✓ Known #30
Technique: common denominator. When do you use it?
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Answer#30
- A difference of fractions leads to \(0/0\).
- Combine into one fraction, simplify, then substitute.
Trap: Using the wrong LCD or canceling before combining.
Algebraic Techniques✓ Known #31
Technique: conjugates. What is the trigger?
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Answer#31
- Radicals give \(0/0\) after substitution.
- Multiply numerator and denominator by the conjugate to use a difference of squares.
Trap: Multiplying only the numerator by the conjugate.
Algebraic Techniques✓ Known #32
Evaluate \(\lim_{x\to 2}\frac{x^2-4}{x-2}\).
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Answer#32
- \(x^2-4=(x-2)(x+2)\)
- Cancel \((x-2)\) ⇒ limit of \(x+2\)
- Substitute: \(2+2=4\)
Trap: Canceling \(x\) with \(x^2\).
Algebraic Techniques✓ Known #33
Evaluate \(\lim_{x\to 0}\frac{\sqrt{x+1}-1}{x}\).
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Answer#33
- Multiply by the conjugate \(\sqrt{x+1}+1\).
- The numerator becomes \(x\), which cancels.
- Limit: \(\frac{1}{\sqrt{x+1}+1}\to \frac12\).
Trap: Skipping the difference-of-squares step.
Algebraic Techniques✓ Known #34
Complex fractions: what is the fastest simplification move?
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Answer#34
- Multiply numerator and denominator by the LCD of the small fractions.
- Simplify, then substitute.
Trap: Multiplying by only one denominator instead of the LCD.
Trig Limits✓ Known #35
What is \(\displaystyle\lim_{x\to 0}\frac{\sin x}{x}\)? (\(x\) in radians)
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Answer#35
- \(\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1\).
- It is proved with the Squeeze Theorem.
Trap: Using degrees, or “canceling” \(x\) from \(\sin x\).
Trig Limits✓ Known #36
What is \(\displaystyle\lim_{x\to 0}\frac{1-\cos x}{x}\)?
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Answer#36
\(0\).
Trap: Mixing it up with \(\frac{\sin x}{x}\), whose limit is 1.
Trig Limits✓ Known #37
Evaluate \(\displaystyle\lim_{x\to 0}\frac{\sin(3x)}{x}\).
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Answer#37
- Rewrite: \(\frac{\sin 3x}{x}=3\cdot\frac{\sin 3x}{3x}\).
- As \(x\to0\), \(3x\to0\), so the limit is \(3\cdot1=3\).
Trap: Answering 1 because it “looks like” \(\frac{\sin x}{x}\).
Continuity✓ Known #38
Continuity at \(x=a\): what are the 3 required conditions?
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Answer#38
- \(f(a)\) exists
- \(\lim_{x\to a} f(x)\) exists
- \(\lim_{x\to a} f(x)=f(a)\)
Trap: Checking the limit but forgetting \(f(a)\).
Continuity✓ Known #39
Continuity of a piecewise function at a breakpoint: what are the exact steps?
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Answer#39
- Compute the left-hand limit from the left piece.
- Compute the right-hand limit from the right piece.
- They must be equal to each other and to \(f(a)\).
Trap: Matching the pieces but forgetting \(f(a)\).
Continuity✓ Known #40
Removable discontinuity: what is the clean definition?
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Answer#40
- The limit exists: \(\lim_{x\to a} f(x)=L\).
- But \(f(a)\) is undefined or \(f(a)\ne L\).
Trap: “The function is undefined ⇒ the limit does not exist.”
Continuity✓ Known #41
Jump discontinuity vs infinite discontinuity: what is the difference?
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Answer#41
- Jump: both one-sided limits are finite but not equal.
- Infinite: at least one one-sided limit is \(\pm\infty\).
Trap: Calling every discontinuity “a hole.”
Continuity✓ Known #42
“Find \(k\) so that \(f\) is continuous at \(x=a\).” What equation determines \(k\)?
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Answer#42
- Compute \(L=\lim_{x\to a} f(x)\) using the rule for \(x\ne a\).
- Set \(k=f(a)=L\).
Trap: Plugging in \(a\) before computing the limit.
Continuity✓ Known #43
If the left and right limits at \(a\) are both 5 and \(f(a)=k\), what must \(k\) be for continuity?
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Answer#43
\(k=5\).
Trap: Thinking matching one-sided limits is enough — that only guarantees the limit exists.
Continuity✓ Known #44
Is every continuous function differentiable?
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Answer#44
No. Differentiability implies continuity, but a continuous function can fail to be differentiable at a corner, cusp or vertical tangent.
Trap: Reversing the implication.
Intermediate Value Theorem✓ Known #45
State the Intermediate Value Theorem (IVT).
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Answer#45
If \(f\) is continuous on \([a,b]\) and \(N\) is between \(f(a)\) and \(f(b)\), then there is at least one \(c\) in \([a,b]\) with \(f(c)=N\).
Trap: Using the IVT without stating that \(f\) is continuous on \([a,b]\).
Intermediate Value Theorem✓ Known #46
What must you show to prove that \(f(x)=0\) has a solution in \((a,b)\)?
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Answer#46
- \(f\) is continuous on \([a,b]\).
- \(f(a)\) and \(f(b)\) have opposite signs.
- Then by the IVT there is \(c\in(a,b)\) with \(f(c)=0\).
Trap: Claiming a root without checking the sign change.
Intermediate Value Theorem✓ Known #47
Does the IVT guarantee a unique solution?
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Answer#47
No. It guarantees at least one solution, not exactly one.
Trap: Writing “exactly one root” from the IVT alone.
Intermediate Value Theorem✓ Known #48
IVT vs approximation: what is the key difference?
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Answer#48
- The IVT proves existence.
- Approximation estimates where the solution is (technology, tables, bisection).
Trap: Mixing existence language with approximation steps.
Graphs & Tables✓ Known #49
From a graph, how do you find \(\lim_{x\to a} f(x)\)?
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Answer#49
- Approach \(a\) from the left and from the right and follow the \(y\)-values.
- If both sides approach the same value \(L\), the limit is \(L\).
- The value \(f(a)\) does not affect the limit.
Trap: Using the filled dot as the limit.
Graphs & Tables✓ Known #50
A graph approaches \(y=3\) at \(x=2\), but the filled dot is \((2,7)\). Find \(\lim_{x\to2}f(x)\) and \(f(2)\).
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Answer#50
- \(\lim_{x\to 2} f(x)=3\)
- \(f(2)=7\)
Trap: Reporting 7 as the limit.
Graphs & Tables✓ Known #51
From a table, how do you estimate \(\lim_{x\to a} f(x)\)?
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Answer#51
- Use \(x\)-values approaching \(a\) from both sides.
- Look for a common trend in the \(y\)-values.
- Report a reasonable estimate.
Trap: Using only one side, which can miss a jump.
Squeeze Theorem✓ Known #52
State the Squeeze Theorem.
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Answer#52
If \(g(x)\le f(x)\le h(x)\) near \(a\) and \(\lim g=\lim h=L\), then \(\lim f=L\).
Trap: Using the theorem without establishing the inequality.
Squeeze Theorem✓ Known #53
If \(-|x|\le f(x)\le |x|\) and \(x\to 0\), what is \(\lim f(x)\)?
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Answer#53
Both bounds approach 0, so \(\lim_{x\to0} f(x)=0\).
Trap: Trying to simplify \(f(x)\) directly when the point is bounding.
Limits at Infinity✓ Known #54
Rational end behavior: what determines \(\lim_{x\to \infty}\frac{P(x)}{Q(x)}\)?
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Answer#54
- Compare the degrees of \(P\) and \(Q\).
- Or divide every term by the highest power of \(x\) in the denominator.
Trap: Plugging in \(\infty\) as if it were a number.
Limits at Infinity✓ Known #55
If \(\deg P<\deg Q\), what is \(\lim_{x\to\infty}\frac{P(x)}{Q(x)}\)?
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Answer#55
\(0\): horizontal asymptote \(y=0\).
Trap: Thinking the graph can never cross the asymptote.
Limits at Infinity✓ Known #56
If \(\deg P=\deg Q\), what is \(\lim_{x\to\infty}\frac{P(x)}{Q(x)}\)?
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Answer#56
The ratio of the leading coefficients: horizontal asymptote \(y=\frac{\text{lead}(P)}{\text{lead}(Q)}\).
Trap: Including lower-degree terms.
Limits at Infinity✓ Known #57
If \(\deg P>\deg Q\), what is \(\lim_{x\to\infty}\frac{P(x)}{Q(x)}\)?
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Answer#57
The limit is \(\pm\infty\) (the sign comes from the leading terms), so there is no horizontal asymptote.
Trap: Writing \(y=0\) or the ratio of leading coefficients.
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