How to prove that a limit with a piece-wise function is false

Question

I am stumped on this practice question that I came across.

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It goes as follows:

$h= 6.62607004\cdot10^{−34}$

$$f(x) = \begin{cases} 0, & \text{if }x < 0 \\ h, & \text{if }x\ge0 \end{cases}$$

Prove that $\lim \limits_{x \to 0}=0$ is false using the epsilon/delta definition of a limit.

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I don't understand how to do this with piece-wise functions. I also have a very difficult time even understanding how to prove limits using the epsilon delta definition.

Answer 1

If $\delta \lt h$, there is no $\epsilon \gt 0$ where $|x-0|\lt \epsilon $ gives $|f(x)-f(0)|\lt \delta$ for all $x$ in the interval.