Calculus I
&
Engineering Mathematics 2
Workshop 2
Sigma Notation
Sigma notation is a compact way to represent sums of many terms:
$\ds \sum_{k=1}^{n} a_k$ $= a_1 + a_2 + a_3 + \cdots + a_n.$
Here, $k$ is the index of summation, $n$ is the upper limit, and $a_k$ is the general term.
Limit (Definition)
$\ds\lim_{x \to a} f(x) = L$
| Formal Definition | Plain-English |
|---|---|
|
Let \(f \colon S \to \R\) be a function and \(c\) a cluster point of \(S \subset \R\). Suppose there exists an \(L \in \R\) and for every \(\epsilon > 0\), there exists a \(\delta \gt 0\) such that whenever \(x \in S \setminus \{ c \}\) and \(|x - c| \lt \delta,\) then \( |f(x) - L| \lt \epsilon \). We say \(L\) is the limit of \(f(x)\) as \(x\) goes to \(c\). |
As \(x\) gets closer to \(c,\) the values of \(f(x)\) get closer and closer to \(L\). |
Continuity
A function $f(x)$ is continuous at $x=a$ if:
$\qquad$ 1. $f(a)$ is defined
$\qquad$ 2. $\ds \lim_{x \to a} f(x)$ exists
$\qquad$ 3. $\ds \lim_{x \to a} f(x) = f(a)$
Summary
| Concept | Notation | Key Idea |
|---|---|---|
| Sigma Notation | $\ds \sum_{k=1}^{n} a_k$ | Compact way to represent a finite sum of terms |
| Limit | $\ds \lim_{x \to a} f(x) = L$ | Describes the value $f(x)$ approaches as $x$ gets close to $a$ |
| Continuity | $\ds \lim_{x \to a} f(x) = f(a)$ | Function has no breaks, jumps, or holes at $x=a$ |