AP - Calculus AB/BC:: Limits and Continuity

What does the statement \(\lim_{x \to a} f(x) = L\) mean, in terms of the behavior of \(f\)?

- **A)** \(f(a) = L\) exactly, regardless of nearby values
- **B)** As \(x\) gets arbitrarily close to \(a\) (from either side, without necessarily equaling \(a\)), \(f(x)\) gets arbitrarily close to \(L\)
- **C)** \(f\) is continuous at \(a\) and equals \(L\) there
- **D)** \(f\) is differentiable at \(a\) with derivative \(L\)

For \(\lim_{x \to a} f(x)\) to exist, which condition must hold?

- **A)** \(f(a)\) must be defined
- **B)** \(f\) must be differentiable at \(a\)
- **C)** The left-hand limit \(\lim_{x \to a^-} f(x)\) and right-hand limit \(\lim_{x \to a^+} f(x)\) must both exist and be equal
- **D)** \(f\) must be a polynomial

Evaluate \(\lim_{x \to 3} (x^2 - 5x + 6)\).

- **A)** \(-6\)
- **B)** \(0\)
- **C)** \(6\)
- **D)** \(3\)

Evaluate \(\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}\).

- **A)** \(0\)
- **B)** The limit does not exist
- **C)** \(4\)
- **D)** \(2\)

Evaluate \(\lim_{x \to 0} \dfrac{\sqrt{x+4} - 2}{x}\).

- **A)** \(\dfrac{1}{4}\)
- **B)** \(0\)
- **C)** \(4\)
- **D)** The limit does not exist

Which limit law justifies \(\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\)?

- **A)** The Squeeze Theorem
- **B)** The Product Law for limits, valid when both individual limits exist
- **C)** The Intermediate Value Theorem
- **D)** L'Hopital's Rule

\(\lim_{x \to 1} \dfrac{x^2 - 1}{x - 1}\) is an example of which indeterminate form before simplification?

- **A)** \(\infty - \infty\)
- **B)** \(1^\infty\)
- **C)** \(0/0\)
- **D)** \(\infty/\infty\)

The Squeeze Theorem states that if \(g(x) \le f(x) \le h(x)\) near \(a\) (except possibly at \(a\)) and \(\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L\), then what can be concluded?

- **A)** \(f\) is continuous at \(a\)
- **B)** \(\lim_{x \to a} f(x) = L\) as well
- **C)** \(f(a) = L\)
- **D)** No conclusion can be drawn about \(f\)