4.3 Some Extensions of the Limit Concept

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Summary

Definition

• f tends to L, x → c+-, resembles limit definition, $x-c,c-x$
• f tends to inf, x → c, resembles limit definition, $f>\alpha ,f<\beta$
• f tends to inf, x → c+, resembles limit definition, $x-c,c-x,f>\alpha ,f<\beta$
• f tends to L, x → inf, resembles limit of sequence
• f tends to inf, x → inf, resembles limit of sequence, $f>\alpha ,f<\beta$

Theorem

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in \mathbb{R}$
• $a\in \mathbb{R}$
• $(a,\infty )\subseteq A$

Then

• $c$ cluster point of $A$
  • $f(x)\le g(x)\forall x\in A,x\ne c$
    • $⟹$ ${\lim }_{x\to c}f=\infty ⟹{\lim }_{x\to c}g=\infty$
    • $⟹$ ${\lim }_{x\to c}g=-\infty ⟹{\lim }_{x\to c}f=-\infty$
• $c\in \mathbb{R}$ cluster point of both $A\cap (c,\infty )$ and $A\cap (-\infty ,c)$
  • ${\lim }_{x\to c}f=L$
    • $⟺{\lim }_{x\to c+}f=L={\lim }_{x\to c-}f$
• $c\in \mathbb{R}$ cluster point of $A\cap (c,\infty )$
  • ${\lim }_{x\to c+}f=L$
    • $⟺$ $\forall (x_n)\in A,(x_n)\to c,x_n>c\forall n\in \mathbb{N}⟹(f(x_n))\to L$
• ${\lim }_{x\to \infty }f=L$
  • $⟺\forall (x_n)\in A\cap (a,\infty )\ni (x_n)\to \infty ⟹(f(x_n))\to L$
• ${\lim }_{x\to \infty }f=\infty$
  • $⟺\forall (x_n)\in A\cap (a,\infty )\ni (x_n)\to \infty ⟹(f(x_n))\to \infty$
• 4.3.15 Theorem

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4.3.1 Definition: f tends to L, x→c+-

Let

• $A\in \mathbb{R}$
• $f:A\to \mathbb{R}$

Then

1. Let

   • $c\in \mathbb{R}$ is a cluster point of $A\cap (c,\infty )=\{x\in A:x>c\}$

   If

   • $\forall \epsilon >0,\exists \delta >0\ni \forall x\in A,0<x-c<\delta ⟹|f(x)-L|<\epsilon$

   Then

   • We say that $L\in \mathbb{R}$ is a right-hand limit of $f$ at $c$
   • ${\lim }_{x\to c+}f=L$

2. Let

   • $c\in \mathbb{R}$ is a cluster point of $A\cap (-\infty ,c)=\{x\in A:x<c\}$

   If

   • $\forall \epsilon >0,\exists \delta >0\ni \forall x\in A,0<c-x<\delta ⟹|f(x)-L|<\epsilon$

   Then

   • We say that $L\in \mathbb{R}$ is a left-hand limit of $f$ at $c$
   • ${\lim }_{x\to c-}f=L$

Resembles definition of limit, but $x-c$ and $c-x$ instead of $|x-c|$

4.3.5 Definition: f tends to inf, x→c

Let

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

Then

1. If

   • $\forall \alpha \in \mathbb{R},\exists \delta >0\ni \forall x\in A,0<|x-c|<\delta ⟹f(x)>\alpha$

   Then

   • We say that $f$ tends to $\infty$ as $x\to c$
   • ${\lim }_{x\to c}f=\infty$

2. If

   • $\forall \beta \in \mathbb{R},\exists \delta >0\ni \forall x\in A,0<|x-c|<\delta ⟹f(x)<\beta$

   Then

   • We say that $f$ tends to $\infty$ as $x\to c$
   • ${\lim }_{x\to c}f=\infty$

Resembles definition of limit, but $f>\alpha$ and $f<\beta$ instead of $|f-L|<\epsilon$

4.3.8 Definition: f tends to inf, x→c+

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in \mathbb{R}$ cluster point of $A\cap (c,\infty )$

If

• $\forall \alpha \in \mathbb{R},\exists \delta >0\ni \forall x\in A,0<x-c<\delta ⟹f(x)>a$ [respectively, $f(x)<\alpha$]

Then

• We say $f$ tends to $\infty$ [respectively, $-\infty$] as $x\to c+$,
• ${\lim }_{x\to c+}f=\infty \left[\text{respectively, }{\lim }_{x\to c+}f=-\infty \right]$

Resembles definition of limit, but combination of 4.3.1 and 4.3.5

4.3.10 Definition: f tends to L, x → inf

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $a\in \mathbb{R}$
• $(a,\infty )\subseteq A$

If

• $\forall \epsilon >0,\exists K>a\ni \forall x>K⟹|f(x)-L|<\epsilon$

Then

• We say that $L\in \mathbb{R}$ is limit of $f$ as $x\to \infty$
• ${\lim }_{x\to \infty }f=L$

Resembles limit of a sequence.

4.3.13 Definition: f tends to inf, x→inf

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $a\in \mathbb{R}$
• $(a,\infty )\subseteq A$

If

• $\forall \alpha \in \mathbb{R},\exists K>a\ni \forall x>K⟹f(x)>\alpha$ [respectively, $f(x)<\alpha$]

Then

• We say $f$ tends to $\infty$ [respectively, $-\infty$] as $x\to \infty$
• ${\lim }_{x\to \infty }f=\infty \left[\text{respectively, }{\lim }_{x\to \infty }f=-\infty \right]$

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4.3.2 Theorem

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in \mathbb{R}$ cluster point of $A\cap (c,\infty )$

Then the following statements are equivalent:

• ${\lim }_{x\to c+}f=L$
• $\forall (x_n)\in A,(x_n)\to c,x_n>c\forall n\in \mathbb{N}⟹(f(x_n))\to L$

TIP

Resembles 4.1.8 Theorem: Sequential criterion, but $x_n>c$, since $x_n$ “approaches” $c$ from the right.

4.3.3 Theorem

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in \mathbb{R}$ cluster point of both $A\cap (c,\infty )$ and $A\cap (-\infty ,c)$

Then ${\lim }_{x\to c}f=L⟺{\lim }_{x\to c+}f=L={\lim }_{x\to c-}f$

No shit bro this is literally 4.1.4 Definition: Limit of a function

4.3.7 Theorem: Squeeze theorem

Let

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

If

• $f(x)\le g(x)\forall x\in A,x\ne c$

Then

1. ${\lim }_{x\to c}f=\infty ⟹{\lim }_{x\to c}g=\infty$
2. ${\lim }_{x\to c}g=-\infty ⟹{\lim }_{x\to c}f=-\infty$

Analogous to 4.2.7 Theorem: Squeeze theorem and 3.6.4 Theorem

4.3.11 Theorem: Sequential criterion for 4.3.10

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $a\in \mathbb{R}$
• $(a,\infty )\subseteq A$

Then the following statements are equivalent:

• ${\lim }_{x\to \infty }f=L$
• $\forall (x_n)\in A\cap (a,\infty )\ni (x_n)\to \infty ⟹(f(x_n))\to L$

As the name suggests, resembles 4.1.8 Theorem: Sequential criterion

4.3.14 Theorem: Sequential criterion for 4.3.13

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $a\in \mathbb{R}$
• $(a,\infty )\subseteq A$

Then the following statements are equivalent:

• ${\lim }_{x\to \infty }f=\infty$ [respectively, ${\lim }_{x\to \infty }f=-\infty$]
• $\forall (x_n)\in (a,\infty )\ni \lim (x_n)=\infty ⟹\lim (f(x_n))=\infty$ [respectively, $\lim (f(x_n))=-\infty$]

4.3.15 Theorem: Ratio test

Let

• $A\subseteq \mathbb{R}$
• $f,g:A\to \mathbb{R}$
• $a\in \mathbb{R}$
• $(a,\infty )\subseteq A$

If

• $g(x)>0\forall x>a$
• ${\lim }_{x\to \infty }\frac{f(x)}{g(x)}=LL\in \mathbb{R},L\ne 0$

Then

• $L>0⟹({\lim }_{x\to \infty }f=\infty ⟺{\lim }_{x\to \infty }g=\infty )$
• $L<0⟹({\lim }_{x\to \infty }f=-\infty ⟺{\lim }_{x\to \infty }g=\infty )$

Analogous to 3.6.5 Theorem: Ratio test