4.2 Limit Theorems

This note covers theorems that are useful in calculating limits of functions. These theorems are parallel to the limit theorems covered in 3.2 Limit Theorems.

<< 4.1 Limits of Function | 4.3 Some Extensions of the Limit Concept >>

Summary

Definition

• 4.2.1 Definition: f bounded on neighborhood of c
• 4.2.3 Definition: Function operations

Theorem

Let

• $b,c\in \mathbb{R}$
• $A\subseteq \mathbb{R}$
• $f,g,h:A\to \mathbb{R}$
• $L,M\in \mathbb{R}$

Then

1. $f$ has a limit at $c$ $⟹$ $f$ bounded on some neighborhood of $c$
2. $c$ cluster point of $A$
   • 4.2.4 Theorem: Algebraic operations on limits
   • 4.2.6 Theorem: Limit between two constants
   • 4.2.7 Theorem: Squeeze theorem
   • 4.2.9 Theorem: Sign preservation

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4.2.1 Definition: f bounded on neighborhood of c

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in \mathbb{R}$ cluster point of $A$.

If

• $\exists V_{\delta }(c),\exists M>0\ni |f(x)|\le M,\forall x\in A\cap V_{\delta }(c)$.

Then

• We say $f$ is bounded on a neigborhood of $c$

Recall

The if block in the definition above is the definition of bounded function.

4.2.3 Definition: Function operations

Let

• $A\subseteq \mathbb{R}$
• $f$ and $g$ be functions defined on $A$ to $\mathbb{R}$.

Then

1. We define the sum $f+g$, the difference $f-g$, and the product $fg$ on $A$ to $\mathbb{R}$ to be the functions given by

$$\begin{array}{rl}(f+g)(x)& :=f(x)+g(x)\\ (f-g)(x)& :=f(x)-g(x)\\ (fg)(x)& :=f(x)g(x)\end{array}$$

for all $x\in A$.

2. If $b\in \mathbb{R}$ we define the multiple $bf$ to be the function given by $(bf)(x):=bf(x)$ for all $x\in A$.

3. If $h(x)\ne 0$ for $x\in A$, we define the quotient $f/h$ to be the function given by $\left(\frac{f}{h}\right)(x):=\frac{f(x)}{h(x)}$ for all $x\in A$.

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4.2.2 Theorem

If

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$ has a limit at $c\in \mathbb{R}$

Then $f$ is bounded on some neighborhood of $c$.

4.2.4 Theorem: Algebraic operations on limits

Let

• $A\subseteq \mathbb{R}$
• $f$ and $g$ be functions on $A$ to $\mathbb{R}$
• $c\in \mathbb{R}$ cluster point of $A$
• $b\in \mathbb{R}$

Then

1. If

   • ${\lim }_{x\to c}f=L$
   • ${\lim }_{x\to c}g=M$

   Then

   $$\begin{array}{rl}\underset{x\to c}{\lim }(f+g)& =L+M\\ \underset{x\to c}{\lim }(f-g)& =L-M\\ \underset{x\to c}{\lim }(fg)& =LM\\ \underset{x\to c}{\lim }(bf)& =bL\end{array}$$

2. If

   • $h:A\to \mathbb{R}$
   • $h(x)\ne 0$ for all $x\in A$
   • ${\lim }_{x\to c}h=H\ne 0$

   Then ${\lim }_{x\to c}\left(\frac{f}{h}\right)=\frac{L}{H}$

4.2.6 Theorem: Limit between two constants

Let

• $A\subseteq \mathbb{R}$
• $f:A\to R$
• $c\in \mathbb{R}$ cluster point of $A$

If

• $a\le f(x)\le b\forall x\in A,x\ne c$
• ${\lim }_{x\to c}f$ exists

Then $a\le {\lim }_{x\to c}f\le b$

4.2.7 Theorem: Squeeze theorem

Let

• $A\subseteq \mathbb{R}$
• $f,g,h:A\to \mathbb{R}$
• $c\in \mathbb{R}$ cluster point of $A$

If

• $f(x)\le g(x)\le h(x)\forall x\in A,x\ne c$
• ${\lim }_{x\to c}f=L={\lim }_{x\to c}h$

Then ${\lim }_{x\to c}g=L$

4.2.9 Theorem: Sign preservation

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in \mathbb{R}$ cluster point of $A$

If

$$\underset{x\to c}{\lim }f>0\left[\text{respectively},\underset{x\to c}{\lim }<0\right]$$

Then

• $\exists V_{\delta }(c)\ni f(x)>0,\forall x\in A\cap V_{\delta }(c),x\ne c\left[\text{respectively},f(x)<0\right]$

NOTE

Basically if $f$ has a limit at $c$ that is positive (or negative), then there exists a neighborhood around $c$, excluding $c$ itself where the function values remain positive (or negative).