Lecture Note: §2 Limits and Continuity
Published:
Last Update: 2025-10-16
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, \(\lim_{x\to c} f(x) = L\)
- 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 \(\lim_{x\to c} f(x) = \lim_{x\to c} g(x)\), provided the limits exists.
- The Squeeze Theorem (The Sandwich Theorem)
§2.3 The Precise Definition of a Limit
- Precise Definition of a Limit, \(\lim_{x\to c} f(x) = L\)
If for every number \(\varepsilon > 0\), there is a number \(\delta > 0\) such that, if \(0 < \lvert x - c\rvert < \delta\) then \(\lvert f(x) - L\rvert < \varepsilon\).
\(\lim_{x\to c} f(x) = L\) 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, \(\lim_{x\rightarrow c^{+}} f(x) = L\)
- 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, \(\lim_{x\rightarrow c^{-}} f(x) = L\)
- If for every number \(\varepsilon > 0\), there is a number \(\delta > 0\) such that, if \(c - \delta < x < c\) then \(\vert f(x) - L\rvert < \varepsilon\).
Theorem: \(\lim_{x \to c } f(x) = L \Longleftrightarrow \lim_{x \to c^{-} } f(x) = L \text{ and } \lim_{x\to c^{+}} f(x) = L \)
Limits at Endpoints of an Interval.
- Two Basic Limits:
- \(\lim_{x \to 0}\dfrac{\sin x}{x} = 1\).
\(\lim_{x \to 0}\dfrac{1 - \cos x}{x} = 0\), or \( \lim_{x \to 0}\dfrac{1 - \cos x}{x^2} = \dfrac{1}{2}\).
§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 \(\lim_{x\to c} f(x) = f(c)\)
- \(f(c)\) is defined (that is, \(c\) is in the domain of \(f\))
- \(\lim_{x\to c} f(x)\) exists
- \(\lim_{x\to c} f(x) = f(c)\)
- 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)
- Properties of Continuous Functions
- Combinations of continuous functions are still continuous.
- Limits of continuous functions:
- If \(g\) is continuous at the point \(b\) and \( \lim_{x \to c} f(x) = b \), then \(\lim_{x \to c} g(f(x)) = g(b) = g\left(\lim_{x \to c} f(x)\right)\).
- Compositions of continuous functions are continuous.
- The Intermediate Value Theorem
- 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 \([ a, b]\).
§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: \(\lim_{x\to \infty} f(x) = L\), \(\lim_{x\to -\infty} f(x) = L\).
- Horizontal asymptote \(y = b\):
- \(\lim_{x \to \infty} f(x) = b \quad \text{or} \quad \lim_{x \to -\infty} f(x) = b\).
- Oblique or slant Asymptote \(y = mx + b\):
- \(\lim_{x \to \infty} \left[f(x) -(mx + b)\right] = 0 \quad \text{or} \quad \lim_{x \to -\infty} \left[f(x) -(mx + b)\right] = 0\).
- Infinite Limits
Definition of infinite limits: \(\lim_{x\to c} f(x) = \infty\), \(\lim_{x\to c} f(x) = -\infty\).
Vertical Asymptotes
Infinite Limits at infinity
- \(\lim_{x\to \infty} f(x) = \infty\), etc.