Example

Use the $\epsilon$ - $\delta$ definition of a limit to prove that:

$\lim\limits_{x\rightarrow c}A=A$ , where

$A$ and

$c$ are real numbers

$\lim\limits_{x\rightarrow c}x=c$ , where

$c$ is a real number

Solution

(a) $f(x)=A$ is the constant function whose graph is a horizontal line. Given any $\epsilon >0$ , we must find $\delta >0$ so that whenever $0 \lt \vert x-c\vert \lt \delta ,$ then $\left\vert f(x)-A\right\vert \lt \epsilon$ .

Since $\left\vert A-A\right\vert =0,$ then $\left\vert f(x)-A\right\vert \lt \epsilon$ no matter what positive number $\delta$ is used. That is, any choice of $\delta$ guarantees that whenever $0 \lt \vert x-c\vert \lt \delta$ , then $\left\vert f(x)-A\right\vert \lt \epsilon$ .

(b) $f( x) =x$ is the identity function. Given any $\epsilon >0$ , we must find $\delta$ so that whenever $0 \lt \vert x-c\vert \lt \delta$ , then $\vert f(x)-c \vert =\vert x-c\vert \lt \epsilon$ . The easiest choice is to make $\delta =\epsilon$ . That is, whenever $0 \lt \vert x-c\vert \lt \delta =\epsilon$ , then $\vert f(x)-c \vert \underset{\underset{\color{#0066A7}{f( x) =x}}{\color{#0066A7}{\uparrow}}}{=} \vert x-c\vert \lt \epsilon$ .