MP 4 :: Definition of Limit

Note: Make sure to use complete sentences when completing the following assignment.

The formal definition of a limit is as follows: the function $f(x)$ has the limit $L$ as $x$ approaches $a$ if for every positive number $\varepsilon$ there exists a number $\delta$ such that if

• $$0<|x-a|<\delta$$

then

• $$|f(x)-L| < \varepsilon .$$

Use this definition to show that the limit of the function

• $$f(x) = \begin{cases} 5 & \text{if } x>0 \\ -1 & \text{if } x \leqslant 0 \end{cases}$$

is not $5$ at $x=0$. In other words, use this definition to show that

• $$\lim_{x\to 0} f(x) \not= 5.$$