Q. 3 - 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 $\delta -\in$definition of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L,\underset{x\to c^{-}}{\lim }f\left(x\right)=L$, and $\underset{x\to c^{+}}{\lim }f\left(x\right)=L$

Step-by-Step Solution

Answer

Limit $\underset{x\to c}{\lim }f\left(x\right)=L$state that if $x\in (c-\delta ,c)\cup (c,c+\delta )$then $f\left(x\right)\in (L-\epsilon ,L+\epsilon ).$ 

Left Limit $\underset{x\to c^{-}}{\lim }f\left(x\right)=L$state that if $x\in (c-\delta ,c)$then $f\left(x\right)\in (L-\epsilon ,L+\epsilon ).$  

Right Limit $\underset{x\to c^{+}}{\lim }f\left(x\right)=L$state that if $x\in (c,c+\delta )$then  $f\left(x\right)\in (L-\epsilon ,L+\epsilon ).$

Step 1. Given information.

The given limits are

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

Step 2. Formal Definition of Limit.

Limit $\underset{x\to c}{\lim }f\left(x\right)=L$state that if $x\in (c-\delta ,c)\cup (c,c+\delta ),$then $f\left(x\right)\in (L-\epsilon ,L+\epsilon )$ so that for all $\epsilon >0,$ there will be $\delta >0.$

 for example in$\underset{x\to 3}{\lim }2x=6$, $x\in (3-\delta ,3)\cup (3,3+\delta )$ and  $2x\in (6-\epsilon ,6+\epsilon )$

Step 3. Formal Definition of Left Limit.

Left Limit $\underset{x\to c}{\lim }f\left(x\right)=L$state that if $x\in (c-\delta ,c),$then $f\left(x\right)\in (L-\epsilon ,L+\epsilon )$ so that for all $\epsilon >0,$ there will be $\delta >0.$

 for example in $\underset{x\to 3^{-}}{\lim }2x=6$, $x\in (3-\delta ,3)$and $2x\in (6-\epsilon ,6+\epsilon )$

Step 4. Formal Definition of Right Limit.

Right Limit $\underset{x\to c}{\lim }f\left(x\right)=L$state that if $x\in (c+\delta ,c),$then $f\left(x\right)\in (L-\epsilon ,L+\epsilon )$ so that for all $\epsilon >0,$ there will be $\delta >0.$

 for example in $\underset{x\to 3^{+}}{\lim }2x=6$, $x\in (3,3+\delta )$and $2x\in (6-\epsilon ,6+\epsilon )$