Chapter 3 — Exercises

42 problems across eight parts. Limits are the foundation every later chapter is built on — practice them seriously. Tier ratings: ⭐ routine recall · ⭐⭐ explain/standard computation · ⭐⭐⭐ harder computation and applications · ⭐⭐⭐⭐ proof and synthesis (math-major track). Selected answers appear in appendices/answers-to-selected.md.

Part A — Conceptual Foundations (⭐)

3.1 Write the intuitive definition of $\lim_{x \to a} f(x) = L$ in your own words, in one sentence, and explain why the value $f(a)$ plays no role (Section 3.2).

3.2 Can $\lim_{x \to a} f(x) = L$ when $f(a)$ is undefined? Give an explicit example and identify what kind of feature the graph has at $a$ (Section 3.2).

3.3 Can $\lim_{x \to a} f(x) \neq f(a)$ when both are defined? Give an example and sketch the graph near $a$ (Section 3.4).

3.4 State, in plain words, the precise condition under which a two-sided limit exists in terms of one-sided limits (Section 3.4).

3.5 Explain the difference between "$\frac{0}{0}$ is indeterminate" and "$\frac{1}{0}$ is undefined." Why must you do algebra in the first case but not the second (Section 3.3)?

3.6 State the Squeeze Theorem in your own words and explain, in one sentence, why a bounded oscillating factor like $\sin(1/x)$ makes it the right tool (Section 3.7).

Part B — Limits by Substitution and the 0/0 Form (⭐⭐)

3.7 Evaluate each limit by direct substitution (Section 3.3): (a) $\lim_{x \to 2} (x^3 - 4x + 1)$ (b) $\lim_{x \to 0} (e^x + 5)$ (c) $\lim_{x \to \pi/4} \cos x$ (d) $\lim_{x \to 1} \ln(2x + 1)$

3.8 Evaluate by factoring and canceling (Section 3.3): (a) $\lim_{x \to 3} \dfrac{x^2 - 9}{x - 3}$ (b) $\lim_{x \to -2} \dfrac{x^2 - 4}{x + 2}$ (c) $\lim_{x \to 1} \dfrac{x^3 - 1}{x - 1}$ (d) $\lim_{x \to 0} \dfrac{x^2 + 3x}{x}$

3.9 Evaluate by rationalizing (Section 3.3): (a) $\lim_{x \to 0} \dfrac{\sqrt{x + 9} - 3}{x}$ (b) $\lim_{x \to 4} \dfrac{\sqrt{x} - 2}{x - 4}$ (c) $\lim_{x \to 1} \dfrac{x - 1}{\sqrt{x} - 1}$

3.10 Evaluate by combining fractions (Section 3.3): (a) $\lim_{x \to 0} \dfrac{\,1/(x+1) - 1\,}{x}$ (b) $\lim_{x \to 2} \dfrac{\,1/x - 1/2\,}{x - 2}$

3.11 These difference quotients are exactly the limits that will define the derivative in Chapter 5. Evaluate each (Section 3.3): (a) $\lim_{h \to 0} \dfrac{(2 + h)^2 - 4}{h}$ (b) $\lim_{h \to 0} \dfrac{(3 + h)^3 - 27}{h}$ (c) $\lim_{h \to 0} \dfrac{\sqrt{4 + h} - 2}{h}$

Part C — One-Sided Limits (⭐⭐)

3.12 For $f(x) = |x|/x$ (Section 3.4): (a) Compute $\lim_{x \to 0^+} f(x)$. (b) Compute $\lim_{x \to 0^-} f(x)$. (c) Does $\lim_{x \to 0} f(x)$ exist? Why?

3.13 For the piecewise function $f(x) = \begin{cases} x^2 + 1 & x < 2 \\ 5 - x & x \geq 2 \end{cases}$ (Section 3.4): (a) Compute $\lim_{x \to 2^-} f(x)$. (b) Compute $\lim_{x \to 2^+} f(x)$. (c) Does $\lim_{x \to 2} f(x)$ exist? What is $f(2)$?

3.14 For $f(x) = \lfloor x \rfloor$ (the greatest integer $\leq x$) (Section 3.4): (a) Compute $\lim_{x \to 1^-} f(x)$. (b) Compute $\lim_{x \to 1^+} f(x)$. (c) Does $\lim_{x \to 1} f(x)$ exist?

3.15 Sketch a single function $f$ satisfying all of: $\lim_{x \to 1^-} f(x) = 3$, $\lim_{x \to 1^+} f(x) = 1$, and $f(1) = 2$. State whether $\lim_{x\to 1} f(x)$ exists (Section 3.4).

Part D — Limits at Infinity (⭐⭐⭐)

3.16 Evaluate (Section 3.5): (a) $\lim_{x \to \infty} \dfrac{3x + 1}{x - 2}$ (b) $\lim_{x \to \infty} \dfrac{x^2 + 5x}{2x^2 - 7}$ (c) $\lim_{x \to \infty} \dfrac{x^2 + 1}{x^3 - 2}$ (d) $\lim_{x \to \infty} \dfrac{x^3 + 2}{x^2 + 5}$ (e) $\lim_{x \to -\infty} \dfrac{4x^2 - 3}{2x^2 + x + 1}$

3.17 $\lim_{x \to \infty} \dfrac{\sqrt{x^2 + 4x}}{x + 2}$. (Hint: divide top and bottom by $x$; since $x > 0$, $\sqrt{x^2} = x$.) (Section 3.5)

3.18 $\lim_{x \to \infty} \left(\sqrt{x^2 + 3x} - x\right)$. (Hint: rationalize by multiplying by the conjugate $\sqrt{x^2+3x}+x$.) (Section 3.5)

3.19 Without computing, state $\lim_{x\to\infty} \dfrac{x^{50}}{e^x}$ and $\lim_{x\to\infty} \dfrac{\ln x}{x}$, and explain in one sentence each which function "wins" (Section 3.5).

Part E — Infinite Limits and Asymptotes (⭐⭐⭐)

3.20 Identify all vertical asymptotes and state the one-sided behavior on each side (Section 3.5): (a) $f(x) = \dfrac{1}{x - 2}$ (b) $g(x) = \dfrac{x + 1}{(x - 1)(x + 3)}$ (c) $h(x) = \tan x$ on $(-\pi/2, \pi/2)$

3.21 Compute, including $\pm\infty$ where appropriate (Section 3.5): (a) $\lim_{x \to 0^+} \ln x$ (b) $\lim_{x \to 0} \dfrac{1}{x^2}$ (c) $\lim_{x \to 0^-} \dfrac{1}{x^3}$ (d) $\lim_{x \to (\pi/2)^-} \tan x$

3.22 A function has horizontal asymptote $y = 4$ as $x \to \infty$ and a vertical asymptote at $x = -1$. Explain, referencing the Common Pitfall in Section 3.5, why these two statements describe geometrically perpendicular features and how the location of "$\infty$" in each limit statement differs.

Part F — The Squeeze Theorem and the Sine Limit (⭐⭐⭐)

3.23 Use the Squeeze Theorem to show $\lim_{x \to 0} x \sin(1/x) = 0$. Begin from $-|x| \leq x \sin(1/x) \leq |x|$ (Section 3.7).

3.24 Use the Squeeze Theorem to show $\lim_{x \to 0} x^2 \cos(1/x) = 0$ (Section 3.7).

3.25 Suppose $1 - x^2 \leq f(x) \leq 1 + x^2$ for all $x$. Find $\lim_{x \to 0} f(x)$, and explain why you can determine it even though $f$ is otherwise unknown (Section 3.7).

3.26 Using the known limit $\lim_{x \to 0} \dfrac{\sin x}{x} = 1$, evaluate (Section 3.7): (a) $\lim_{x \to 0} \dfrac{\sin 3x}{x}$ (b) $\lim_{x \to 0} \dfrac{\sin 5x}{\sin 2x}$ (c) $\lim_{x \to 0} \dfrac{1 - \cos x}{x^2}$ (Hint: $1 - \cos x = 2\sin^2(x/2)$.) (d) $\lim_{x \to 0} \dfrac{\tan x}{x}$

Part G — Applications (⭐⭐⭐)

3.27 (Physics — instantaneous velocity.) A ball dropped from rest falls $s(t) = 4.9 t^2$ meters in $t$ seconds. The average velocity over $[1, 1+h]$ is $\dfrac{s(1+h) - s(1)}{h}$. Compute $\lim_{h\to 0}$ of this quotient to find the instantaneous velocity at $t = 1$ s (Sections 3.3, 3.11).

3.28 (Economics — marginal cost.) A firm's cost to produce $q$ units is $C(q) = 0.01 q^2 + 5q + 200$ dollars. The marginal cost at $q = 100$ is $\lim_{h\to 0} \dfrac{C(100 + h) - C(100)}{h}$. Evaluate it and state its units (Sections 3.3, 3.11).

3.29 (Biology — saturating growth.) A population follows $P(t) = \dfrac{500\,t}{t + 4}$ (in individuals, $t$ in days). Find $\lim_{t\to\infty} P(t)$ and interpret it as a carrying capacity (Section 3.5).

3.30 (Data science — learning-rate decay.) A training schedule uses step size $\eta(t) = \dfrac{0.5}{1 + 0.1t}$ at iteration $t$. Compute $\lim_{t\to\infty}\eta(t)$ and explain in one sentence why a vanishing step size helps gradient descent settle (Section 3.5).

3.31 (Economics — tax-bracket step.) A marginal tax rate is $r(x) = 0.12$ for income $x < 44{,}725$ and $r(x) = 0.22$ for $x \ge 44{,}725$ (dollars). Compute $\lim_{x\to 44725^-} r(x)$ and $\lim_{x\to 44725^+} r(x)$, and report the size of the jump (Section 3.4).

Part H — Computational with Python (⭐⭐⭐)

3.32 Use sympy to verify your answers to 3.8, 3.9, and 3.16 (Section 3.9):

import sympy as sp
x = sp.symbols('x')
print(sp.limit((x**2 - 9)/(x - 3), x, 3))           # check 3.8(a)
print(sp.limit((sp.sqrt(x) - 2)/(x - 4), x, 4))      # check 3.9(b)
print(sp.limit((x**2 + 5*x)/(2*x**2 - 7), x, sp.oo)) # check 3.16(b)

3.33 Use sympy's directional argument to compute one-sided limits and confirm your hand answer to 3.12 (Section 3.9):

print(sp.limit(sp.Abs(x)/x, x, 0, '+'))   # right-hand
print(sp.limit(sp.Abs(x)/x, x, 0, '-'))   # left-hand

3.34 Use numpy to build a numerical table for $\lim_{x\to 0}\dfrac{\sin x}{x}$ at $x = 10^{-1}, 10^{-2}, 10^{-3}, 10^{-4}$, and confirm the values march toward $1$. Then explain why the table alone does not prove the limit (Section 3.9).

3.35 Use matplotlib to plot $f(x) = \sin(1/x)$ on $(0,\,0.1]$ and describe, in two sentences, what the picture shows about why $\lim_{x\to 0^+}\sin(1/x)$ does not exist (Section 3.2).

Part I — ε-δ Proofs and Synthesis (⭐⭐⭐⭐, math-major track)

3.36 Prove $\lim_{x \to 1} (3x + 2) = 5$ using the ε-δ definition. Show your scratch work finding $\delta(\varepsilon)$, then write the clean forward proof (Section 3.10).

3.37 Prove $\lim_{x \to a} c = c$ for any constants $a, c$ directly from the ε-δ definition. (Hint: $|f(x) - L| = 0 < \varepsilon$ for every $x$, so any $\delta$ works.) (Section 3.10)

3.38 Prove $\lim_{x \to 2} x^2 = 4$ using the ε-δ definition. (Hint: $|x^2 - 4| = |x-2|\,|x+2|$; first require $\delta \le 1$ so that $|x+2| < 5$, then take $\delta = \min\{1,\ \varepsilon/5\}$.) (Section 3.10)

3.39 Prove $\lim_{x \to 3} (x^2 - x) = 6$ using the ε-δ definition. (Hint: factor $|x^2 - x - 6| = |x-3|\,|x+2|$ and bound $|x+2|$ by restricting $\delta \le 1$.) (Section 3.10)

3.40 Prove that a limit, if it exists, is unique. Suppose $\lim_{x\to a} f(x) = L_1$ and $\lim_{x\to a} f(x) = L_2$; assume $L_1 \neq L_2$ and derive a contradiction by taking $\varepsilon = |L_1 - L_2|/2$ (Section 3.10).

3.41 Prove the Constant-Multiple Limit Law: if $\lim_{x\to a} f(x) = L$, then $\lim_{x\to a} c\,f(x) = cL$. Handle the case $c = 0$ separately, then for $c \neq 0$ use $\delta$ chosen against $\varepsilon/|c|$ (Sections 3.6, 3.10).

3.42 Using the ε-δ box picture (Section 3.10), explain in a short paragraph why $\lim_{x\to 0}\,\text{sgn}(x)$ fails to exist: no single $L$ allows a vertical strip around $0$ to keep the graph inside a horizontal band of half-width $\varepsilon = \tfrac12$. Connect this to the one-sided-limit diagnostic of Section 3.4.

Difficulty Distribution

Total: 42 exercises, roughly 5–6 hours of focused work. Application problems span physics, economics, biology, and data science (Part G). Selected answers are in appendices/answers-to-selected.md.