Q. 4 - Calculus

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 

Step-by-Step Solution

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 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 be$\delta >0$.

 For example in $\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 $x\in (N,\infty )$then $f\left(x\right)\in (L-\epsilon ,L+\epsilon )$ so that for all $\epsilon >0,$ there will be $N>0.$

 For example in $\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 $x\in (N,\infty )$then $f\left(x\right)\in (M,\infty ).$ so that for all $M>0,$ there will be $N>0.$

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