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

The given functions with limits are

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

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$2.$

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

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.