Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

the formal$\delta -M,N-\in ,$, and N–M definitions of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=\infty ,\underset{x\to \infty }{\lim }f\left(x\right)=L,$and$\underset{x\to \infty }{\lim }f\left(x\right)=\infty$, respectively

Short answer

Infinite limit $\underset{x\to c}{\lim }f\left(x\right)=\infty$state that if $x\in (c-\delta ,c)\cup (c,c+\delta )$then $f\left(x\right)\in (M,0).$

Limit at the infinite $\underset{x\to \infty }{\lim }f\left(x\right)=L$state that if $x\in (N,0),$then $f\left(x\right)\in (L-\epsilon ,L+\epsilon ).$

Infinity limit at infinite $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$state that if $x\in (N,\infty )$then $f\left(x\right)\in (M,0).$

Step-by-step solution

Step 1. Given information. 

The given limits are

$$\begin{gathered} \underset{x\to c}{\lim }f\left(x\right)=\infty \\ \underset{x\to \infty }{\lim }f\left(x\right)=L \\ \underset{x\to \infty }{\lim }f\left(x\right)=\infty \end{gathered}$$

Step 2. Formal Definition of infinite limit. 

Infinite limit $\underset{x\to c}{\lim }f\left(x\right)=\infty$state that if $x\in (c-\delta ,c)\cup (c,c+\delta )$then$f\left(x\right)\in (M,\infty ).$so that for all $M>0,$there will belocalid="1648273818000" $\delta >0$.

For example in localid="1648273876726" $\underset{x\to 1}{\lim }\frac{1}{x+1}=\infty ,$$x\in (1-\delta ,1)\cup (1,1+\delta )$and$\frac{1}{x+1}\in (M,\infty )$.

Step 3. Formal Definition of limit at the infinite. 

Limit at the infinite $\underset{x\to \infty }{\lim }f\left(x\right)=L$state that if localid="1648273312084" $x\in (N,\infty )$then $f\left(x\right)\in (L-\epsilon ,L+\epsilon )$so that for all localid="1648273281300" $\epsilon >0,$there will be $N>0.$

For example in localid="1648274034934" $\underset{x\to \infty }{\lim }\frac{1}{x}+1=1,$$x\in (N,\infty ),$and $\frac{1}{x}+1\in (1-\epsilon ,1+\epsilon ).$

Step 4. Formal Definition of infinity limit at infinite. 

Infinity limit at infinite $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$state that if localid="1648274119796" $x\in (N,\infty )$then $f\left(x\right)\in (M,\infty ).$so that for all localid="1648273404018" $M>0,$there will be localid="1648273395865" $N>0.$

For example in$\underset{x\to \infty }{\lim }x=\infty ,$$x\in (N,\infty )$and $f\left(x\right)\in \infty$.