Dr. Xiaozhou Li

Dr. Xiaozhou Li

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Lecture Note: §2 Limits and Continuity

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Last Update: 2026-10-13

In this chapter, we develop the concept of a limit, first intuitively and then formally. We use limits to describe the way a function varies.

§2.1 Rates of Change and Tangent Lines to Curves

  • The Velocity Problem
    • Find the instantaneous velocity of an object moving along a straight path at a specific time if the position of the object at any time is known.
    • Average velocity, instantaneous velocity
  • The Tangent Problem
    • We can think of a tangent to a curve as a line that touches the curve and follows the same direction as the curve at the point of contact.
    • The slope of the tangent line
    • Two points, secant line

§2.2 Limit of a Function and Limit Laws

  • Intuitive Definition of a Limit, $\displaystyle \lim_{x\to c} f(x) = L$
    • The limit of $f(x)$, as $x$ approaches $c$, equals $L$.
    • If we can make the values of $f(x)$ arbitrarily close to $L$ (as close to $L$ as we like) by restricting $x$ to be sufficiently close to $c$ (on either side of $c$) but not equal to $c$.
    • We do not consider what happens when $x$ is actually equals $c$!
  • Limit Laws
    • Constant function, identity function
    • It is easy to believe that these rules are true: Sum, Difference, Constant Multiple, Product, Quotient, Power, Root Rules.
    • Simplify limit computations.
  • Direct Substitution
    • Limits of Polynomials
    • Limits of Rational Functions
      • Eliminating Common Factors from Zero Denominators:
      • If $f(x) = g(x)$ when $x\neq c$, then $\displaystyle \lim_{x\to c} f(x) = \lim_{x\to c} g(x)$, provided the limits exists.
  • The Squeeze Theorem (The Sandwich Theorem)

§2.3 The Precise Definition of a Limit

  • Precise Definition of a Limit, $\displaystyle \lim_{x\to c} f(x) = L$

    • If for every number $\varepsilon > 0$, there is a number $\delta > 0$ such that, if $0 < \lvert x - c\rvert < \delta$ then $\lvert f(x) - L\rvert < \varepsilon$.

    • $\displaystyle \lim_{x\to c} f(x) = L$ means that

      • The distance between $f(x)$ and $L$ can be made arbitrarily small by requiring that the distance from $x$ to $a$ be sufficiently small (but not $0$).
      • The values of $f(x)$ can be made as close as we please to $L$ by requiring $x$ to be close enough to $c$ (but not equal to $c$).
      • For every $\varepsilon >0$ (no matter how small $\varepsilon$ is) we can find $\delta > 0$ such that if $x$ lies in the open interval $(c-\delta, c+\delta)$ and $x \neq c$, then $f(x)$ lies in the open interval $(L-\varepsilon, L +\varepsilon)$.

  • How to find algebraically a $\delta$ for a given $f, L, c$, and $\varepsilon >0$. (How to use the $\varepsilon - \delta$ language).

  • Using the Definition to Prove Theorems (Limit Laws, The Squeeze Theorem, etc.).

§2.4 One-Sided Limits

We extend the limit concept to one-sided limits — approaching a limit from one side.

  • Precise Definition of Right-Hand Limit, $\displaystyle \lim_{x\rightarrow c^{+}} f(x) = L$
    • If for every number $\varepsilon > 0$, there is a number $\delta > 0$ such that, if $c < x < c+\delta$ then $\lvert f(x) - L\rvert < \varepsilon$.
  • Precise Definition of Left-Hand Limit, $\displaystyle \lim_{x\rightarrow c^{-}} f(x) = L$
    • If for every number $\varepsilon > 0$, there is a number $\delta > 0$ such that, if $c - \delta < x < c$ then $\lvert f(x) - L\rvert < \varepsilon$.

  • Theorem: $\displaystyle \lim_{x \to c } f(x) = L \Longleftrightarrow \lim_{x \to c^{-} } f(x) = L \text{ and } \lim_{x\to c^{+}} f(x) = L$

  • Limits at Endpoints of an Interval.

  • Two Basic Limits:
    • $\displaystyle\lim_{x \to 0}\dfrac{\sin x}{x} = 1$.

    • $\displaystyle\lim_{x \to 0}\dfrac{1 - \cos x}{x} = 0$, or $\displaystyle\lim_{x \to 0}\dfrac{1 - \cos x}{x^2} = \dfrac{1}{2}$.

§2.5 Continuity

Intuitively, any function $y = f (x)$ whose graph can be sketched over its domain in one unbroken motion is an example of a continuous function.

  • Definition: A function $f$ is continuous at a number $c$ if $\displaystyle\lim_{x\to c} f(x) = f(c)$
    • $f(c)$ is defined (that is, $c$ is in the domain of $f$)
    • $\displaystyle\lim_{x\to c} f(x)$ exists
    • $\displaystyle\lim_{x\to c} f(x) = f(c)$
  • Three common types of discontinuities:
    • Removable discontinuity: $\lim\limits_{x\to c} f(x)$ exists, but either the limit does not equal $f(c)$ or $f(c)$ is not defined.
    • Jump discontinuity: The one-sided limits both exist but are not equal.
    • Infinite discontinuity: The limit is infinite as $x$ approaches $c$ from one or both sides.
  • Continuous function
    • If $f$ is continuous at every point in its domain, then we say that $f$ is continuous (or $f$ is a continuous function)

  • Laws of Continuity: Sums, products, multiples, inverses, and composites of continuous functions are continuous. The same holds for a quotient $f/g$ at points where $g(x) \neq 0$.

  • Limits of continuous functions:
    • If $g$ is continuous at the point $b$ and $\displaystyle\lim_{x \to c} f(x) = b$, then $\displaystyle\lim_{x \to c} g(f(x)) = g(b) = g\left(\lim_{x \to c} f(x)\right)$.

  • Compositions of continuous functions are continuous.

  • Substitution Method: If $f$ is known to be continuous at $x = c$, then the value of the limit $\lim\limits_{x\to c} f(x)$ is $f(c)$.

  • Indeterminate Form
    • If the formula for $f(c)$ yields an undefined expression of the type \(\frac{0}{0}, \frac{\infty}{\infty}, \infty\cdot 0, \infty - \infty\) then we say that $f(x)$ is indeterminate (or has an indeterminate form) at $x = c$.
    • Try to transform $f(x)$ algebraically into a new expression that is defined and continuous at $x = c$, and then evaluate by substitution.
  • Intermediate Value Theorem (IVT)
    • The IVT says that a continuous function cannot skip values.
    • If $f$ is a continuous function on a closed interval $[a,b]$, and if $y_0$ is any value between $f(a)$ and $f(b)$, then $y_0 = f (c)$ for some $c$ in $[a, b]$.
    • Existence of zeros: If $f$ is continuous on $[a, b]$, and if one of $f(a)$ and $f(b)$ is nonnegative and the other is nonpositive, then $f$ has a zero in $[a, b]$.

§2.6 Limits Involving Infinity; Asymptotes of Graphs

In this section, we investigate the behavior of a function when the magnitude of the independent variable $x$ becomes increasingly large, or $x \to \pm \infty$. We further extend the concept of limit to infinite limits. Infinite limits provide useful symbols and language for describing the behavior of functions whose values become arbitrarily large in magnitude.

  • Limits at infinity (finite limits as $x \to \pm \infty$)

    • The symbol for infinity ($\infty$) does not represent a real number. We use it to describe the behavior of a function when the values in its domain or range outgrow all finite bounds.

    • Definitions of limits at infinity: $\displaystyle\lim_{x\to \infty} f(x) = L$, $\displaystyle\lim_{x\to -\infty} f(x) = L$.

    • Horizontal asymptote $y = b$:
      • $\displaystyle\lim_{x \to \infty} f(x) = b \quad \text{or} \quad \lim_{x \to -\infty} f(x) = b$.
    • Oblique or slant Asymptote $y = mx + b$:
      • $\displaystyle\lim_{x \to \infty} \left[f(x) -(mx + b)\right] = 0 \quad \text{or} \quad \lim_{x \to -\infty} \left[f(x) -(mx + b)\right] = 0$.
      • Horizontal asymptote is the special case of oblique asymptote ($m = 0$).
  • Infinite Limits

    • Definition of infinite limits: $\displaystyle\lim_{x\to c} f(x) = \infty$, $\displaystyle\lim_{x\to c} f(x) = -\infty$.

    • Vertical Asymptotes

    • Infinite Limits at infinity

      • $\displaystyle\lim_{x\to \infty} f(x) = \infty$, etc.
  • Conceptual insight of dominant terms.