Calculus Bible Chapter 1: Functions and Limits

This post summarizes Chapter 1 of James Stewart's Calculus.

1.1 Definition and Representations of a Function#

Function: A function $f$ is a rule that assigns to each element $x$ in a set $D$ exactly one element, called $f(x)$, in a set $E$. (p.3)

We consider functions where both $D$ and $E$ are sets of real numbers.

• Domain: the set $D$
• Range: the set of all possible values of $f(x)$ as $x$ varies throughout the domain (p.3)

Functions are classified into two types based on symmetry. (p.8)

• Even function: satisfies $f(-x) = f(x)$ for every $x$ in its domain (symmetric about the y-axis)
• Odd function: satisfies $f(-x) = -f(x)$ for every $x$ in its domain (symmetric about the origin)

1.2 Essential Functions#

Mathematical model: A mathematical representation of a real-world phenomenon — such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, a person's life expectancy at birth, or the cost of reducing exhaust emissions. The purpose of a mathematical model is to understand the phenomenon and to make predictions about future behavior. (p.12)

Major types of functions (p.12)

• Linear function: a function whose graph is a straight line
• Polynomial function: a function with multiple terms
• Reciprocal function: a hyperbolic function with equation $y = 1/x$ or $xy = 1$, having the coordinate axes as asymptotes ($f(x) = x^{-1} = 1/x$)
• Rational function: a function expressed as the ratio of two polynomials

$f(x) = \frac{P(x)}{Q(x)}$

Transformations of Functions (p.19)

Transformations include vertical/horizontal shifts, vertical/horizontal stretches, and reflections.

Combinations of Functions (p.21)

Given functions $f$ and $g$, the composite function $f \circ g$ is defined as follows.

$(f \circ g)(x) = f(g(x))$

1.3 The Limit of a Function#

Definition of a Limit (p.27)#

Suppose $f(x)$ is defined when $x$ is near the number $a$. (This means $f$ is defined on some open interval that contains $a$, except possibly at $a$ itself.)

If we can make the values of $f(x)$ arbitrarily close to $L$ by taking $x$ to be sufficiently close to $a$ (on either side of $a$) but not equal to $a$, then we write:

$\lim_{x \to a} f(x) = L$

This is read as "the limit of $f(x)$ as $x$ approaches $a$ equals $L$."

Definition of One-Sided Limits (p.32)#

If we can make the values of $f(x)$ arbitrarily close to $L$ by taking $x$ to be sufficiently close to $a$ with $x$ less than $a$, then we write:

$\lim_{x \to a^-} f(x) = L$

"The left-hand limit of $f(x)$ as $x$ approaches $a$ is $L$." (The opposite is the right-hand limit.)

Necessary and Sufficient Condition for a Limit

$\lim\limits_{x \to a} f(x) = L$ if and only if $\lim\limits_{x \to a^-} f(x) = L$ and $\lim\limits_{x \to a^+} f(x) = L$.

1.4 Calculating Limits#

The following laws are used for computing limits. (p.37)

\lim_{x \to a} c &= c \quad \text{(Constant Law)} \\[0.3em] \lim_{x \to a} x &= a \quad \text{(Identity Law)} \\[0.3em] \lim_{x \to a} [c \cdot f(x)] &= c \cdot \lim_{x \to a} f(x) \quad \text{(Constant Multiple Law)} \\[0.3em] \lim_{x \to a} [f(x) \pm g(x)] &= \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) \quad \text{(Sum/Difference Law)} \\[0.3em] \lim_{x \to a} [f(x) \cdot g(x)] &= \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \quad \text{(Product Law)} \\[0.3em] \lim_{x \to a} \frac{f(x)}{g(x)} &= \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \quad \text{(Quotient Law, } \lim_{x \to a} g(x) \neq 0) \\[0.3em] \lim_{x \to a} [f(x)]^n &= \left[\lim_{x \to a} f(x)\right]^n \quad \text{(Power Law)} \\[0.3em] \lim_{x \to a} \sqrt[n]{f(x)} &= \sqrt[n]{\lim_{x \to a} f(x)} \quad \text{(Root Law)} \\[0.3em] \lim_{x \to a} f(g(x)) &= f\!\left(\lim_{x \to a} g(x)\right) \quad \text{(Composite Function Law, } f \text{ continuous)} \end{aligned}$$ <br /> ## 1.5 Continuity ### Definition of Continuity (p.48) A function $f$ is called **continuous at** $a$ if the following holds. <br /> $$\lim_{x \to a} f(x) = f(a)$$ <br /> This definition implicitly requires three conditions. 1. $f(a)$ is defined (i.e., $a$ is in the domain of $f$). 2. $\lim\limits_{x \to a} f(x)$ exists. 3. $\lim\limits_{x \to a} f(x) = f(a)$ Types of discontinuity (p.50) - Removable discontinuity - Infinite discontinuity - Jump discontinuity **One-Sided Continuity** (p.51) A function $f$ is **continuous from the right at** $a$ if the following holds. <br /> $$\lim_{x \to a^+} f(x) = f(a)$$ <br /> A function $f$ is **continuous from the left at** $a$ if the following holds. <br /> $$\lim_{x \to a^-} f(x) = f(a)$$ <br /> ## 1.6 Limits Involving Infinity ### Infinite Limits (p.60) > In general, the following notation means that the values of $f(x)$ tend to grow larger and larger as $x$ approaches $a$. It means we can make $f(x)$ as large as we like by taking $x$ sufficiently close to $a$ (on either side of $a$) but not equal to $a$. <br /> $$\lim_{x \to a} f(x) = \infty$$ <br /> **Vertical Asymptotes** (p.61) The line $x = a$ is called a **vertical asymptote** of the curve $y = f(x)$ if at least one of the following statements is true. - $\lim\limits_{x \to a} f(x) = \infty$ - $\lim\limits_{x \to a^-} f(x) = \infty$ - $\lim\limits_{x \to a^+} f(x) = \infty$ - $\lim\limits_{x \to a} f(x) = -\infty$ - $\lim\limits_{x \to a^-} f(x) = -\infty$ - $\lim\limits_{x \to a^+} f(x) = -\infty$ ### Limits at Infinity (p.63) > Suppose $f$ is defined on some interval $(a, \infty)$. > The following notation means that we can make $f(x)$ as close to $L$ as we like by taking $x$ sufficiently large. <br /> $$\lim_{x \to \infty} f(x) = L$$ <br /> **Horizontal Asymptotes** (p.64) The line $y = L$ is called a **horizontal asymptote** of the curve $y = f(x)$ if either of the following holds. <br /> $$\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L$$ <br /> If $n$ is a positive integer, then the following hold. (p.66) <br /> $$\lim_{x \to \infty} \frac{1}{x^n} = 0, \quad \lim_{x \to -\infty} \frac{1}{x^n} = 0$$ <br /> ### Infinite Limits at Infinity (p.68) The following notation is used to indicate that $f(x)$ becomes large as $x$ becomes large. <br /> $$\lim_{x \to \infty} f(x) = \infty$$ <br /> Similarly, the following notation is used. <br /> $$\lim_{x \to -\infty} f(x) = \infty, \quad \lim_{x \to \infty} f(x) = -\infty, \quad \lim_{x \to -\infty} f(x) = -\infty$$ <br /> ### Precise Definitions **Precise Definition of an Infinite Limit** (p.69) Suppose $f$ is defined on some open interval containing $a$ (except possibly at $a$ itself). Then $\lim\limits_{x \to a} f(x) = \infty$ means that for every positive number $M$, there exists a $\delta$ such that: <br /> $$0 < |x - a| < \delta \implies f(x) > M$$ <br /> **Precise Definition of a Limit at Infinity** (p.70) Suppose $f$ is defined on some interval $(a, \infty)$. Then $\lim\limits_{x \to \infty} f(x) = L$ means that for every $\epsilon > 0$ there is a number $N$ such that if $x > N$ then $|f(x) - L| < \epsilon$. **Precise Definition of an Infinite Limit at Infinity** (p.71) Suppose $f$ is defined on some interval $(a, \infty)$. Then $\lim\limits_{x \to \infty} f(x) = \infty$ means that for every positive number $M$ there is a positive number $N$ such that if $x > N$ then $f(x) > M$. <br /> ## Summary The topics covered in Chapter 1 form the conceptual foundation for all of calculus. - **Functions**: domain and range, symmetry of even and odd functions - **Limits**: a limit exists when both the left-hand and right-hand limits are equal - **Continuity**: a function is continuous when the limit value equals the function value - **Limits involving infinity**: vertical and horizontal asymptotes reveal the overall behavior of a function