Infinite Limits and Sequences

Limit at Infinity

$$\begin{equation*} \lim_{x \to +\infty} f(x) = A \end{equation*}$$

means that

$$\begin{equation*} \forall \varepsilon > 0,\; \exists M > 0 \text{ such that } \forall x \in I,\; x > M \Rightarrow |f(x) - A| < \varepsilon. \end{equation*}$$

Similarly,

$$\begin{equation*} \lim_{x \to -\infty} f(x) = A \end{equation*}$$

means

$$\begin{equation*} \forall \varepsilon > 0,\; \exists M > 0 \text{ such that } \forall x \in I,\; x < -M \Rightarrow |f(x) - A| < \varepsilon. \end{equation*}$$ $$\begin{equation*} \lim_{x \to \infty} f(x) = A \end{equation*}$$

means

$$\begin{equation*} \forall \varepsilon > 0,\; \exists M > 0 \text{ such that } \forall x \in I,\; |x| > M \Rightarrow |f(x) - A| < \varepsilon. \end{equation*}$$

Deleted Neighborhoods

Let $c$ be one of $x_0,\; x_0^+,\; x_0^-,\; -\infty,\; +\infty,\; \infty$. The neighborhood and deleted neighborhood of $c$ are defined as follows:

$c$	Neighborhood of $c$	Deleted neighborhood of $c$
$x_0$	An open interval $J$ containing $x_0$	$J \setminus \{x_0\}$
$x_0^+$	$(x_0,\, x_0+\delta)$	$(x_0,\, x_0+\delta)$
$x_0^-$	$(x_0-\delta,\, x_0)$	$(x_0-\delta,\, x_0)$
$-\infty$	$(-\infty,\, b)$	$(-\infty,\, b)$
$+\infty$	$(a,\, +\infty)$	$(a,\, +\infty)$
$\infty$	$(-\infty,\, b)\,\cup\,(a,\, +\infty)$	$(-\infty,\, b)\,\cup\,(a,\, +\infty)$

General Definition of Limits

Let $c$ be one of $x_0,\; x_0^+,\; x_0^-,\; -\infty,\; +\infty,\; \infty$, and let $A$ be a real number or one of $-\infty,\; +\infty,\; \infty$.

We say that $\lim_{x \to c} f(x) = A$ iff:

For every neighborhood $V$ of $A$, there exists a deleted neighborhood $W$ of $c$ such that for all $x \in I \cap W$, we have $f(x) \in V$.

Infinite Quantities

When $x \to c$, if $f(x)$ grows without bound, we say $f(x)$ is an infinite quantity (or positive/negative infinite quantity).

If

$$\begin{equation*} \lim_{x \to c} f(x) = \infty \quad \text{(i.e. } \lim_{x \to c} f(x) = +\infty \text{ or } \lim_{x \to c} f(x) = -\infty \text{)}, \end{equation*}$$

then $f(x)$ diverges to infinity as $x$ approaches $c$.

If

$$\begin{equation*} \lim_{x \to c} f(x) = A \in \{-\infty, +\infty, \infty\}, \qquad \lim_{y \to A} g(y) = B, \end{equation*}$$

then

$$\begin{equation*} \lim_{x \to c} g(f(x)) = B. \end{equation*}$$

However, if

$$\begin{equation*} \lim_{x \to c} f(x) = A, \qquad \lim_{y \to A} g(y) = B \in \{-\infty, +\infty, \infty\}, \end{equation*}$$

then

$$\begin{equation*} \lim_{x \to c} g(f(x)) = B \end{equation*}$$

is not necessarily true.

Sequences and Their Limits

A sequence $\{a_n\}_{n \ge 1}$ can be regarded as a function defined on the set of positive integers $\mathbb{N}^*$. Hence, its limit can be defined as

$$\begin{equation*} \lim_{n \to +\infty} a_n. \end{equation*}$$

Theorems

1. Let $\{n_k\}_{k \ge 1}$ be a strictly increasing sequence of positive integers. Define $b_k = a_{n_k}$. Then $\{b_k\}_{k \ge 1}$ is called a subsequence of $\{a_n\}_{n \ge 1}$.

If

$$\begin{equation*} \lim_{n \to +\infty} a_n = A \in \mathbb{R} \cup \{-\infty, +\infty, \infty\}, \end{equation*}$$

then

$$\begin{equation*} \lim_{k \to +\infty} a_{n_k} = \lim_{k \to + \infty}b_k = A. \end{equation*}$$

2. Let $f:(a,b) \to \mathbb{R}$ be a monotone non-decreasing function.

   • If $f$ is bounded above, then

     $$\begin{equation*} \lim_{x \to b^-} f(x) = \sup_{x \in (a,b)} f(x). \end{equation*}$$

   • If $f$ is unbounded above, then

     $$\begin{equation*} \lim_{x \to b^-} f(x) = +\infty. \end{equation*}$$

3. Let $\{a_n\}$ be a monotone non-decreasing sequence.

   • If $\{a_n\}$ is bounded above, then

     $$\begin{equation*} \lim_{n \to +\infty} a_n = \sup_{n \ge 1} a_n. \end{equation*}$$

   • If $\{a_n\}$ is unbounded above, then

     $$\begin{equation*} \lim_{n \to +\infty} a_n = +\infty. \end{equation*}$$

Definition of e

Let two sequences be defined as

$$\begin{equation*} a_n = \left(1 + \frac{1}{n}\right)^n, \quad b_n = \left(1 + \frac{1}{n}\right)^{n+1}. \end{equation*}$$

Then:

• $a_n$ is monotonically increasing.
• $b_n$ is monotonically decreasing.
• For all $n$, $a_n < b_n$.
• Therefore, both sequences converge and have the same limit.

We define this common limit as

$$\begin{equation*} e = \lim_{n \to +\infty} \left(1 + \frac{1}{n}\right)^n. \end{equation*}$$

From Discrete to Continuous

$$\begin{equation*} \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = e. \end{equation*}$$

Theorems

1. $\lim\limits_{x \to 0} \frac{\ln(1+x)}{x} = 1$.
2. $\lim\limits_{x \to 0} \frac{\sin x}{x} = 1$.
3. $\lim\limits_{x \to 0} \frac{a^x - 1}{x} = \ln a$.
4. $\lim\limits_{x \to 1} \frac{x^\mu - 1}{x - 1} = \mu$.