Notation
This is the notation used to write "the limit of f(x) as x approaches c."
lim ₓ→c f(x)
On a graph with an open circle (hole) at x = c, how do you read off lim ₓ→c f(x)?
It's the y-value the hole sits at — trace both sides toward that height, even though f(c) is undefined.
Given lim f(x) = L and lim g(x) = M as x→c, state the limit law for lim [f(x) + g(x)].
lim [f(x) + g(x)] = L + M
What technique should you reach for on lim ₓ→2 (x² − 4)/(x − 2)?
Factor the numerator and cancel the common (x − 2) factor.
If direct substitution gives 0/0, what does that tell you?
It's an indeterminate form. NOT proof the limit is 0 or DNE. It means more algebraic work is needed.
True or false: for lim ₓ→c f(x) to exist, f(c) itself must be defined.
False
A table shows f(x) → 7 as x → 2 from both the left (1.9, 1.99, 1.999) and the right (2.1, 2.01, 2.001). What is lim ₓ→2 f(x)?
7
State the limit law for lim [f(x) · g(x)] as x→c.
lim [f(x) · g(x)] = L · M
A limit with a square root gives 0/0, like lim ₓ→0 (√(x+4) − 2)/x. What technique clears it?
Rationalize OR L'Hôpital's Rule
Name 3 different representations a limit problem can appear in on the AP exam.
Possible answers include graphically, numerically (tables), analytically (algebraically), or verbally.
For a two-sided limit to exist at x = c, these two one-sided limits must be equal.
The left-hand limit (x→c⁻) and the right-hand limit (x→c⁺)
At a vertical asymptote x = 3, f(x) → ∞ from the left and f(x) → −∞ from the right. What is lim ₓ→3 f(x)?
DNE
What extra condition do you need before applying the quotient law lim [f(x)/g(x)] = L/M?
M ≠ 0
State the Squeeze Theorem.
If g(x) ≤ f(x) ≤ h(x) near c (except possibly at c), and lim g(x) = lim h(x) = L as x→c, then lim f(x) = L too.
On the AP exam, what exactly must you state to justify that a limit does not exist because the one-sided limits disagree?
Both one-sided limit values and a statement that they are unequal
If lim ₓ→c⁻ f(x) = 3 and lim ₓ→c⁺ f(x) = 5, what do you conclude about lim ₓ→c f(x)?
DNE
A graph has a jump at x = 1: the curve approaches y = 2 from the left, y = 4 from the right, and there's a filled dot at (1, 4). Give lim ₓ→1 f(x) and f(1).
The limit DNE (2 ≠ 4); f(1) = 4.
If lim ₓ→c f(x) = L, what is lim ₓ→c [f(x)]ⁿ?
Lⁿ
Evaluate: lim ₓ→3 (x² − 9)/(x − 3)
6
True or false: if f is undefined at x = c, the limit as x→c automatically does not exist.
False
Explain the difference between f(c) and lim ₓ→c f(x), and give a situation where they differ.
f(c) is the actual function value at c; the limit is the value f(x) approaches near c. They can differ at a removable discontinuity (a hole), where f(c) is undefined or reassigned.
A scientist records a chemical's temperature only from sensor readings as time approaches t = 4 seconds — there's no formula for the reaction. Why is a table the only way to estimate lim ₜ→4 T(t) here, and what does this method NOT guarantee?
Since there's no algebraic formula for T(t), techniques like substitution, factoring, or L'Hôpital's Rule can't be used at all.
A table of recorded values is the only option. But it only estimates and can't confirm that the data keeps the same trend beyond the points actually measured.
Why can't you always evaluate lim ₓ→c f(x) by simply plugging x = c into the function?
Direct substitution only works where f is continuous at c.
Use squeeze-theorem reasoning to explain why lim ₓ→0 x²·sin(1/x) = 0.
Since −1 ≤ sin(1/x) ≤ 1, multiplying by x² gives −x² ≤ x²sin(1/x) ≤ x². Both bounds → 0 as x→0, so by the Squeeze Theorem the limit is 0.
Final question: f(x) = (x² − 1)/(x − 1) for x ≠ 1, and f(1) = 5. Find lim ₓ→1 f(x), and state whether f is continuous at x = 1.
2; not continuous