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AP Calculus Flash Cards Unit 1
AP Calculus AB Flashcards
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Unit 1 — Limits & Continuity
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Limit Laws
When is it valid to evaluate a limit by
direct substitution
?
Answer
If the function is
continuous
at the point (or the expression stays defined).
Polynomials are continuous everywhere; rational functions are continuous where denominator \(\neq 0\).
Trap:
Substituting into a rational expression where the denominator becomes \(0\).
Limit Laws
If \(\lim_{x\to a} f(x)=L\) and \(\lim_{x\to a} g(x)=M\), what is \(\lim_{x\to a} [f(x)+g(x)]\)?
Answer
\(\lim_{x\to a}[f(x)+g(x)]=L+M\).
Trap:
Applying laws when one of the limits does not exist.
Limit Laws
What is \(\lim_{x\to a} [c\cdot f(x)]\) if \(c\) is a constant and \(\lim_{x\to a} f(x)=L\)?
Answer
\(\lim_{x\to a} [c f(x)] = cL\).
Trap:
Forgetting to distribute constants properly.
Limit Laws
State the product and quotient limit laws (when they apply).
Answer
\(\lim (fg) = (\lim f)(\lim g)\)
\(\lim \dfrac{f}{g}=\dfrac{\lim f}{\lim g}\) if \(\lim g\neq 0\)
Trap:
Using quotient law when \(\lim g=0\).
Limit Laws
If \(\lim_{x\to a} f(x)=L\), what is \(\lim_{x\to a} [f(x)]^n\) (integer \(n\))?
Answer
\(\lim [f(x)]^n = L^n\).
Trap:
Confusing \([f(x)]^n\) with \(f(nx)\).
Limit Laws
If \(\lim_{x\to a} f(x)=L\) and \(f(x)\ge 0\) near \(a\), what is \(\lim_{x\to a} \sqrt{f(x)}\)?
Answer
\(\lim \sqrt{f(x)}=\sqrt{L}\), assuming \(L\ge 0\).
Trap:
Taking \(\sqrt{L}\) when \(L<0\) (not real).
Limit Laws
What is \(\lim_{x\to a} |f(x)|\) if \(\lim_{x\to a} f(x)=L\)?
Answer
\(\lim |f(x)| = |L|\).
Trap:
Dropping absolute value without considering sign.
Limit Laws
Why do we often rewrite expressions before taking a limit?
Answer
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
Define the
left-hand
and
right-hand
limits at \(x=a\).
Answer
Left: \(\lim_{x\to a^-} f(x)\)
Right: \(\lim_{x\to a^+} f(x)\)
Trap:
Thinking \(a^-\) means “negative a.”
One-Sided Limits
When does the two-sided limit \(\lim_{x\to a} f(x)\) exist?
Answer
\(\lim_{x\to a^-} f(x)\) exists
\(\lim_{x\to a^+} f(x)\) exists
They are
equal
Trap:
“Both exist ⇒ limit exists.” (Must be equal.)
One-Sided Limits
What does a
jump discontinuity
look like in limit language?
Answer
\(\lim_{x\to a^-} f(x)\neq \lim_{x\to a^+} f(x)\).
Trap:
Writing the two-sided limit anyway.
One-Sided Limits
Piecewise at \(x=a\): what should you compute first?
Answer
Compute \(\lim_{x\to a^-} f(x)\) using the left piece.
Compute \(\lim_{x\to a^+} f(x)\) using the right piece.
Then compare; check \(f(a)\) separately.
Trap:
Plugging \(a\) into the wrong piece.
One-Sided Limits
If \(\lim_{x\to a^-} f(x)=\infty\), what does that mean?
Answer
As \(x\to a^-\), \(f(x)\) grows without bound (vertical-asymptote behavior from the left).
Trap:
Treating \(\infty\) as a number.
One-Sided Limits
Quick check: \(\lim_{x\to 0^-} \frac{|x|}{x}\) and \(\lim_{x\to 0^+} \frac{|x|}{x}\)?
Answer
\(x\to 0^-\): \(|x|=-x\Rightarrow \frac{|x|}{x}=-1\)
\(x\to 0^+\): \(|x|=x\Rightarrow \frac{|x|}{x}=1\)
Trap:
Treating \(|x|\) as always \(x\).
Infinite Limits
What does \(\lim_{x\to a} f(x)=\infty\) indicate?
Answer
Unbounded behavior near \(x=a\) (often a vertical asymptote).
Trap:
Assuming it’s two-sided without checking both one-sided limits.
Infinite Limits
If left limit \(\to\infty\) and right limit \(\to-\infty\), does the two-sided limit exist?
Answer
No. The two-sided limit does not exist (different one-sided behaviors).
Trap:
“Both infinite ⇒ exists.” Not necessarily.
Infinite Limits
Compare: \(\lim_{x\to 0}\frac{1}{x}\) vs \(\lim_{x\to 0}\frac{1}{x^2}\).
Answer
\(\frac{1}{x}\): left \(\to -\infty\), right \(\to \infty\) ⇒ two-sided DNE.
\(\frac{1}{x^2}\): both sides \(\to \infty\).
Trap:
Forgetting sign changes when power is odd.
Infinite Limits
How to distinguish a
hole
vs a
vertical asymptote
in a rational function?
Answer
Factor and cancel common factors first.
If a factor cancels ⇒ removable discontinuity (hole).
If denominator still \(=0\) after simplification ⇒ vertical asymptote behavior.
Trap:
Declaring an asymptote before simplifying.
Infinite Limits
If \(f(x)=\frac{1}{(x-a)^3}\), what happens as \(x\to a^-\) and \(x\to a^+\)?
Answer
\(x\to a^-\): \((x-a)^3<0\Rightarrow f(x)\to -\infty\)
\(x\to a^+\): \((x-a)^3>0\Rightarrow f(x)\to \infty\)
Trap:
Treating odd powers like even powers.
Infinite Limits
If \(\lim_{x\to a^+} f(x)=\infty\), what should you write (AP language)?
Answer
“As \(x\) approaches \(a\) from the right, \(f(x)\) increases without bound.”
Trap:
Writing \(\infty\) as a “value” instead of behavior.
Indeterminate Forms
You get \(\frac{0}{0}\). What should you try
first
in AB?
Answer
Factor & cancel
Common denominator
Conjugate (radicals)
Trap:
Writing “\(0/0\)” as the answer.
Indeterminate Forms
\(\frac{\infty}{\infty}\): what does it mean?
Answer
Indeterminate. The limit could be \(0\), finite nonzero, or \(\pm\infty\). You must rewrite/simplify.
Trap:
“Infinity over infinity equals 1.”
Indeterminate Forms
Is \(\frac{1}{0}\) indeterminate?
Answer
No. It indicates unbounded behavior (infinite limit/vertical asymptote), depending on one-sided signs.
Trap:
Calling every denominator-zero case “0/0.”
Indeterminate Forms
If \(\lim f=0\) and \(\lim g=0\), what can you conclude about \(\lim \frac{f}{g}\)?
Answer
Nothing definite—\(\frac{0}{0}\) is indeterminate. You must rewrite/simplify.
Trap:
Concluding the limit is 1 because “both go to 0.”
Indeterminate Forms
If a factor \((x-a)\) cancels, what does that usually mean about the original function at \(x=a\)?
Answer
The original function has a
removable discontinuity
(a hole) at \(x=a\).
The limit may still exist even if \(f(a)\) is undefined.
Trap:
Calling it a vertical asymptote before canceling.
Indeterminate Forms
Goal of rewriting an indeterminate form?
Answer
Find an equivalent expression (for \(x\neq a\)) where substitution works.
Or identify one-sided unbounded behavior.
Trap:
Doing non-equivalent algebra (changing the function).
Indeterminate Forms
Not indeterminate: is \(\frac{0}{5}\) indeterminate? What is the value (if valid)?
Answer
Not indeterminate. If valid, \(\frac{0}{5}=0\).
Trap:
Overusing “indeterminate” whenever you see a zero.
Indeterminate Forms
Not indeterminate: is \(\frac{\infty}{5}\) indeterminate? What is the behavior?
Answer
Not indeterminate. It grows without bound (behavior \(\to\infty\)).
Trap:
Treating \(\infty\) as a cancelable number.
Algebraic Techniques
Technique:
Factoring
. Typical trigger?
Answer
If substitution gives \(0/0\) and you see a polynomial/difference, factor to cancel a common factor (often \(x-a\)).
Trap:
Canceling terms that are being added (only cancel common factors).
Algebraic Techniques
Technique:
Common denominator
. When?
Answer
Difference of fractions leads to \(0/0\).
Combine into one fraction, simplify, then substitute.
Trap:
Incorrect LCD or canceling before combining.
Algebraic Techniques
Technique:
Conjugates
. Trigger?
Answer
Radicals with \(0/0\) after substitution.
Multiply by conjugate to use difference of squares.
Trap:
Multiplying by the conjugate on only the numerator.
Algebraic Techniques
Evaluate: \(\lim_{x\to 2}\frac{x^2-4}{x-2}\).
Answer
\(x^2-4=(x-2)(x+2)\)
Cancel \((x-2)\Rightarrow\) limit of \(x+2\)
Substitute: \(2+2=4\)
Trap:
Canceling \(x\) with \(x^2\) (illegal).
Algebraic Techniques
Evaluate: \(\lim_{x\to 0}\frac{\sqrt{x+1}-1}{x}\).
Answer
Multiply by conjugate \(\sqrt{x+1}+1\).
Numerator becomes \(x\), cancels with denominator.
Limit: \(\frac{1}{\sqrt{x+1}+1}\to \frac12\).
Trap:
Forgetting the difference of squares step.
Algebraic Techniques
Complex fractions: what’s the fastest AB simplification move?
Answer
Multiply numerator and denominator by the
LCD
of the small fractions.
Simplify, then substitute.
Trap:
Multiplying by only one denominator instead of the LCD.
Continuity
Continuity at \(x=a\): what are the
3 required conditions
?
Answer
\(f(a)\) 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
Piecewise continuity at a breakpoint: what are the exact steps?
Answer
Compute left-hand limit from left piece.
Compute right-hand limit from right piece.
Set them equal (for limit to exist) and equal to \(f(a)\).
Trap:
Setting only the pieces equal and forgetting \(f(a)\).
Continuity
Removable discontinuity: what’s the clean definition?
Answer
The limit exists: \(\lim_{x\to a} f(x)=L\).
But \(f(a)\) is undefined or \(f(a)\ne L\).
Trap:
“Function not defined ⇒ limit DNE.”
Continuity
Jump discontinuity vs infinite discontinuity: what’s the difference?
Answer
Jump:
one-sided limits exist but are not equal.
Infinite:
at least one one-sided limit is \(\pm\infty\).
Trap:
Calling any discontinuity “a hole.”
Continuity
“Find \(k\) so \(f\) is continuous at \(x=a\).” What equation determines \(k\)?
Answer
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
If left and right limits are both 5 but \(f(a)=k\), what must \(k\) be for continuity?
Answer
\(k=5\).
Trap:
Thinking matching one-sided limits is enough (that only guarantees the limit exists).
Continuity
Is every continuous function differentiable? (AB concept check)
Answer
No. Differentiability implies continuity, but a continuous function can fail to be differentiable (corner/cusp/vertical tangent).
Trap:
Reversing the implication.
IVT
State the
Intermediate Value Theorem
(IVT).
Answer
If \(f\) is continuous on \([a,b]\) and \(N\) is between \(f(a)\) and \(f(b)\), then \(\exists c\in(a,b)\) such that \(f(c)=N\).
Trap:
Using IVT without stating continuity on \([a,b]\).
IVT
Root existence template: what must you show to prove \(f(x)=0\) has a solution in \((a,b)\)?
Answer
\(f\) is continuous on \([a,b]\).
\(f(a)\) and \(f(b)\) have opposite signs.
Then IVT ⇒ \(\exists c\in(a,b)\) with \(f(c)=0\).
Trap:
Claiming a root without checking the sign change.
IVT
Does IVT guarantee a
unique
solution?
Answer
No. IVT guarantees
at least one
solution, not uniqueness.
Trap:
Writing “exactly one root” from IVT alone.
IVT
IVT vs approximation: what’s the key difference?
Answer
IVT: proves existence.
Approximation: estimates where the solution is (bisection/technology/etc.).
Trap:
Mixing “existence proof” language with approximation steps.
Graphs & Tables
From a graph, how do you find \(\lim_{x\to a} f(x)\)?
Answer
Approach \(a\) from left and right and track the y-values.
If both sides approach the same value \(L\), then the limit is \(L\).
Ignore \(f(a)\) unless asked.
Trap:
Using the filled dot value as the limit.
Graphs & Tables
If the graph shows approach value 3 at \(x=2\), but the filled dot is \((2,7)\), what are \(\lim_{x\to2}f(x)\) and \(f(2)\)?
Answer
\(\lim_{x\to 2} f(x)=3\)
\(f(2)=7\)
Trap:
Reporting 7 as the limit.
Graphs & Tables
From a table, what’s the correct AP approach to estimate \(\lim_{x\to a} f(x)\)?
Answer
Use x-values approaching \(a\) from both sides.
Look for a common y-value trend.
Report a reasonable estimate (often rounded).
Trap:
Using only one side (can miss a jump).
Squeeze Theorem
State the
Squeeze Theorem
.
Answer
If \(g(x)\le f(x)\le h(x)\) near \(a\) and \(\lim g=\lim h=L\), then \(\lim f=L\).
Trap:
Using Squeeze without establishing the inequality.
Squeeze Theorem
If \(-|x|\le f(x)\le |x|\) and \(x\to 0\), what is \(\lim f(x)\)?
Answer
Both bounds go to 0, so \(\lim f(x)=0\).
Trap:
Trying to simplify \(f(x)\) directly when the whole point is bounding.
Limits at Infinity
Rational end behavior: what determines \(\lim_{x\to \infty}\frac{P(x)}{Q(x)}\)?
Answer
Compare degrees of \(P\) and \(Q\).
Divide by the highest power of \(x\) in the denominator.
Trap:
Plugging \(\infty\) in as if it were a number.
Limits at Infinity
If \(\deg(P)<\deg(Q)\), what is \(\lim_{x\to\infty}\frac{P(x)}{Q(x)}\)?
Answer
0 (horizontal asymptote \(y=0\)).
Trap:
Thinking it “reaches” 0 at a finite x-value.
Limits at Infinity
If \(\deg(P)=\deg(Q)\), what is \(\lim_{x\to\infty}\frac{P(x)}{Q(x)}\)?
Answer
Ratio of leading coefficients (horizontal asymptote \(y=\frac{\text{lead}(P)}{\text{lead}(Q)}\)).
Trap:
Including lower-degree terms.
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