Chapter 3 — Self-Assessment Quiz

10 questions, ~20 minutes. Each answer cites the section it tests. Aim for 7/10 before moving on.

1. $\lim_{x \to 2} f(x) = L$ means:

  • A) $f(2) = L$
  • B) $f(x)$ gets arbitrarily close to $L$ as $x$ gets close to $2$ (but $x \neq 2$)
  • C) $L$ is the maximum value of $f$ near $x = 2$
  • D) $f(x) = L$ for all $x$ in some interval around $2$
Answer **B)** The limit captures *approach*, not value. $f(2)$ might or might not equal $L$; the function need not even be defined at $2$. *Reference: Section 3.2.*

2. $\lim_{x \to 1} \dfrac{x^2 - 1}{x - 1} = $

  • A) $0$
  • B) $1$
  • C) $2$
  • D) Undefined (the function isn't defined at $x = 1$)
Answer **C) $2$.** Factor: $(x-1)(x+1)/(x-1) = x + 1$ for $x \neq 1$, so the limit is $2$. The value at $x = 1$ is irrelevant. *Reference: Section 3.2.*

3. When direct substitution into a quotient gives $0/0$, the correct next move is to:

  • A) Conclude the limit doesn't exist
  • B) Set the limit equal to $0$
  • C) Factor, rationalize, or combine fractions to remove the offending factor, then substitute again
  • D) Set the limit equal to $\infty$
Answer **C)** $0/0$ is *indeterminate*: the limit may be any number, $\infty$, or DNE. Algebra must resolve the race between numerator and denominator. *Reference: Section 3.3.*

4. $\lim_{x \to 0^+} \dfrac{1}{x} = $

  • A) $0$
  • B) $1$
  • C) $+\infty$
  • D) Does not exist as a finite number, but we write $+\infty$ (a vertical asymptote)
Answer **D)** As $x \to 0^+$, $1/x$ grows past every bound. We write $\lim_{x\to0^+} 1/x = +\infty$ as shorthand for a blow-up; $\infty$ is not a number. (Choice C is also commonly accepted; D is the more careful statement.) *Reference: Section 3.5.*

5. For $f(x) = \begin{cases} x + 1 & x \leq 0 \\ x^2 - 1 & x > 0 \end{cases}$, $\lim_{x \to 0} f(x)$ is:

  • A) $-1$ (the right-hand limit)
  • B) $1$ (the left-hand limit)
  • C) Does not exist (the one-sided limits disagree)
  • D) $0$
Answer **C)** Left: $x + 1 \to 1$. Right: $x^2 - 1 \to -1$. The one-sided limits disagree, so the two-sided limit does not exist. *Reference: Section 3.4.*

6. $\lim_{x \to \infty} \dfrac{3x^2 + 7}{x^2 + 5x + 1} = $

  • A) $0$
  • B) $3$
  • C) $\infty$
  • D) $7$
Answer **B) $3$.** Numerator and denominator have equal degree ($2$), so the limit is the ratio of leading coefficients, $3/1 = 3$. *Reference: Section 3.5.*

7. $\lim_{x \to 0} \dfrac{\sin x}{x} = $

  • A) $0$
  • B) $1$
  • C) Does not exist
  • D) $\infty$
Answer **B) $1$.** The keystone trigonometric limit, proved by a geometric squeeze and used to differentiate $\sin x$ in Chapter 7. *Reference: Section 3.7.*

8. Which limit is best evaluated using the Squeeze Theorem?

  • A) $\lim_{x \to \infty} \dfrac{1}{x}$
  • B) $\lim_{x \to 0} x^2 \sin(1/x)$
  • C) $\lim_{x \to 0} \dfrac{1}{x}$
  • D) $\lim_{x \to 1} (x^2 + 1)$
Answer **B)** Since $-x^2 \leq x^2\sin(1/x) \leq x^2$ and both bounds tend to $0$, the squeeze forces the limit to $0$. The oscillating factor makes direct methods useless. *Reference: Section 3.7.*

9. $\lim_{x \to 0} \dfrac{\sqrt{x + 4} - 2}{x} = $

  • A) $0$
  • B) $1/2$
  • C) $1/4$
  • D) Does not exist
Answer **C) $1/4$.** Rationalize: $\dfrac{\sqrt{x+4}-2}{x}\cdot\dfrac{\sqrt{x+4}+2}{\sqrt{x+4}+2} = \dfrac{1}{\sqrt{x+4}+2} \to \dfrac{1}{4}$. *Reference: Section 3.3.*

10. In the formal ε-δ definition of the limit, the role of $\delta$ is:

  • A) An upper bound on $|f(x) - L|$ (the output distance)
  • B) An upper bound on $|x - a|$ (the input distance)
  • C) The limit value $L$ itself
  • D) The point $a$ being approached
Answer **B)** $\delta$ controls how close $x$ must be to $a$; $\varepsilon$ controls how close $f(x)$ must be to $L$. The definition guarantees: for every output tolerance $\varepsilon$, some input tolerance $\delta$ achieves it. *Reference: Section 3.10.*

Scoring

  • 9–10: Solid mastery. Continue to Chapter 4 (Continuity).
  • 7–8: Good. Skim any section you missed and move on.
  • 5–6: Re-read Sections 3.3 (the 0/0 form) and 3.4–3.5 (one-sided limits and infinity). Redo Parts B and D of the exercises.
  • Below 5: Re-read the whole chapter slowly, then watch the 3Blue1Brown video on limits alongside it. Retake the quiz.