Chapter 3 — Key Takeaways

The Big Ideas

1. A limit captures behavior, not value. $\lim_{x \to a} f(x) = L$ means $f(x)$ approaches $L$ as $x$ approaches $a$, with $x \neq a$. The function need not be defined at $a$, and even if it is, the limit may differ from $f(a)$ (Section 3.2).

2. A limit can fail for three distinct reasons. Diagnose why, not just that it fails: a jump (one-sided limits disagree, e.g. $\text{sgn}(x)$ at $0$), a blow-up (the function runs off to $\pm\infty$, e.g. $1/x$ at $0$), or oscillation (the function never settles, e.g. $\sin(1/x)$ at $0$). Being undefined at the point is never by itself a reason — the hole case $\frac{x^2-1}{x-1}$ is undefined at $1$ yet has a perfectly good limit (Section 3.2).

3. For nice functions at nice points, limit = value. $\lim_{x \to a} f(x) = f(a)$ for polynomials, exponentials, logs, and sines/cosines wherever they are defined. This "plug in" rule is the definition of continuity, made precise in Chapter 4 (Section 3.3).

4. The $0/0$ form is indeterminate. It is a promise more work is needed, not a verdict — never "$=0$" or "$=1$." Resolve it by algebra: factor and cancel, rationalize with the conjugate, or combine fractions; failing that, defer to L'Hôpital's Rule in Chapter 9. Distinguish it from $\frac10$, which is genuinely undefined and signals a blow-up (Section 3.3).

5. One-sided limits and the existence rule. $\lim_{x \to a^+}$ approaches from the right, $\lim_{x \to a^-}$ from the left. The two-sided limit exists if and only if both one-sided limits exist and are equal. This is the precise diagnostic for the jump failure (Section 3.4).

6. Two different infinities. Limits at infinity ($x \to \pm\infty$) describe long-run behavior and give horizontal asymptotes. Infinite limits ($f \to \pm\infty$ at a finite $a$) describe blow-ups and give vertical asymptotes. They are geometrically perpendicular; read the location of "$\infty$" carefully (Section 3.5).

7. The rational-function rule at infinity. For $p(x)/q(x)$ as $x\to\pm\infty$: if $\deg p < \deg q$, the limit is $0$; if $\deg p = \deg q$, it is the ratio of leading coefficients; if $\deg p > \deg q$, it is $\pm\infty$. Technique: divide top and bottom by the highest power in the denominator (Section 3.5).

8. The limit laws. When $\lim f = L$ and $\lim g = M$ both exist finitely, limits pass through sums, differences, constant multiples, products, quotients (if $M \neq 0$), powers, and roots. The finiteness hypothesis is essential: with infinities the laws can break, producing indeterminate forms $0\cdot\infty$, $\infty - \infty$, $\infty/\infty$, $0/0$ that the laws cannot resolve (Section 3.6).

9. The Squeeze Theorem. If $g(x) \le f(x) \le h(x)$ near $a$ and $g, h$ both tend to $L$, then $f \to L$ too. The workhorse for functions you cannot attack directly but can bound — especially a tame factor (like $x^2$) times a wild bounded one (like $\sin(1/x)$) (Section 3.7).

10. The ε-δ definition (math-major track, Section 3.10): $\lim_{x\to a} f(x) = L$ means for every $\varepsilon > 0$ there exists $\delta > 0$ with $0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon$. It does not replace the intuitive picture — it certifies it, recasting "gets close" as a challenge-and-response game in finite tolerances. Optional for engineers and scientists; essential for analysis.

Limit Laws (Quick Reference)

Assume $\lim_{x\to a} f = L$ and $\lim_{x\to a} g = M$ both finite.

Built on the two atoms $\lim_{x\to a} c = c$ and $\lim_{x\to a} x = a$, these laws prove the substitution rule for polynomials.

Standard Limits to Memorize

• $\lim_{x \to a} c = c$ and $\lim_{x \to a} x = a$
• $\lim_{x \to 0} \dfrac{\sin x}{x} = 1$ (keystone — powers the derivative of $\sin$ in Chapter 7)
• $\lim_{x \to 0} \dfrac{1 - \cos x}{x} = 0$ and $\lim_{x\to 0}\dfrac{1-\cos x}{x^2} = \dfrac12$
• $\lim_{x \to 0} \dfrac{\tan x}{x} = 1$
• $\lim_{x \to \infty} \dfrac{1}{x} = 0$
• $\lim_{x \to \infty} \dfrac{p(x)}{q(x)} = $ ratio of leading coefficients (when $\deg p = \deg q$)
• $\lim_{x \to \infty} \dfrac{x^n}{e^x} = 0$ for any $n$ (exponential beats any polynomial)
• $\lim_{x \to \infty} \dfrac{\ln x}{x^n} = 0$ for any $n > 0$ (log loses to any positive power)

The Strategy, in One Glance (Section 3.8)

1. Substitute. Finite and nice? Done.
2. Get $0/0$? Factor, rationalize, or combine fractions; substitute again.
3. $x \to \pm\infty$? Divide by the highest power, or compare growth rates.
4. Bounded oscillating factor? Squeeze.
5. Piecewise, $|x|$, suspected vertical asymptote? Compute one-sided limits and compare.
6. Indeterminate form survives? Defer to L'Hôpital (Chapter 9).

Skills You Should Now Have

• Evaluate any limit of a continuous function by direct substitution.
• Recognize and resolve $0/0$ by factoring, rationalizing, or combining fractions.
• Compute one-sided limits and decide whether the two-sided limit exists.
• Evaluate limits at infinity for rational functions in one line.
• Apply the Squeeze Theorem when bounds are available.
• Use the limit laws, knowing exactly when their hypotheses fail.
• (Math majors) Read and produce a basic ε-δ proof, including the $\delta = \min\{1, \varepsilon/K\}$ technique.
• Use numpy to gather numerical evidence and sympy to certify a limit symbolically.

Common Pitfalls

• Limit ≠ value. Do not assume $\lim_{x\to a} f(x) = f(a)$ without justification; substitution is valid only for continuous functions.
• $0/0$ is not $0$ (or $1$, or anything). It is indeterminate — the answer could be any number, $\pm\infty$, or DNE.
• Two infinities confused. $\lim_{x\to\infty} f$ (horizontal asymptote) is not $\lim_{x\to a} f = \infty$ (vertical asymptote).
• Limit laws applied to infinities. The laws require finite limits; "$\infty - \infty = 0$" is false.
• Radians, not degrees. Every trig limit (especially $\frac{\sin x}{x}\to 1$) assumes radian measure.
• (ε-δ) $\delta$ must not depend on $x$. It may depend on $\varepsilon$ and $a$, but it is a single number good for the whole strip; if a stray $x$ appears, bound it first by restricting $\delta \le 1$.

Connections Across the Book

• Continuity (Chapter 4) is defined by a limit: $f$ is continuous at $a$ iff $\lim_{x\to a} f(x) = f(a)$.
• The derivative (Chapters 5–6) is the limit $f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$ — exactly the difference quotient of Case Study 2.
• The definite integral (Chapter 13) is a limit of Riemann sums.
• Series convergence (Chapters 20–22) is the limit of partial sums; the Squeeze Theorem reappears as the comparison test.
• The Fundamental Theorem of Calculus (Chapter 14) ties the two limit-built operations — derivative and integral — together.

A Reflection

The limit is the single most important concept in calculus, and the hardest to truly internalize — many students compute limits fluently long before they grasp what a limit is. Take your time, and re-read this chapter as the idea returns. The first time you use a limit to build a derivative (Chapter 5), then an integral (Chapter 13), then a convergent series (Chapter 21), the concept sharpens.

If you carry away two sentences, carry these: limits are about approach, not arrival, and every concept in calculus is built from limits. With those, you have the foundation for everything that follows.