Q. 1 - 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 intuitive meaning 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: If f is a function and c and L are real numbers where the value of $f\left(x\right)$reaches L as x reaches c then we can say that L is the limit of $f\left(x\right)$as x approaches c.

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

Left-hand limit: If f is a function and c and L are real numbers where the value of $f\left(x\right)$reaches L as x reaches c from left then we can say that L is the left-hand limit of$f\left(x\right)$as x approaches c. 

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

Right-hand Limit: If f is a function and c and L are real numbers where the value of $f\left(x\right)$ reaches L as x reaches c  from right then we can say that L is the right-hand limit of $f\left(x\right)$as x approaches c. 

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

Step 1. Given information.

The given functions with 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. Description of limit.

Limit: If f is a function and c and L are real numbers where the value of$f\left(x\right)$ reaches L as x reaches c then we can say that L is the limit of $f\left(x\right)$as x approaches c.

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

For example in$\underset{x\to 2}{\lim }f\left(x\right)=1$, function value reaches to $1$ as x reaches to $2.$

Step 3. Description of left-hand limit.

Left-hand limit: If f is a function and c and L are real numbers where the value of $f\left(x\right)$reaches L as x reaches c from left then we can say that L is the left-hand limit of$f\left(x\right)$as x approaches c.

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

For example in$\underset{x\to 2^{-}}{\lim }f\left(x\right)=1$, function value reaches to $1$ as x reaches to $2$ from left determines

Step 4. Description of Right-hand limit.

Right-hand Limit: If f is a function and c and L are real numbers where the value of $f\left(x\right)$ reaches L as x reaches c  from right then we can say that L is the right hand limit of $f\left(x\right)$as x approaches c.

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

For example in$\underset{x\to -2^{+}}{\lim }f\left(x\right)=1$, function value reaches to $1$ as x reaches to $-2$ from right determines.