2. Delta & Epsilon

$$\lim_{x \to a} f(x) = L$$

we make $f(x)$ as close to $L$ as we can by forcing $x$ to be close enough to $a$.

 

Delta ($\delta$)

$\delta$ is a positive number that specifies how close $x$ must be to $a$ so that $f(x)$ is sufficiently close to $L$

$$0 < |x - a| < \delta$$

Epsilon ($\epsilon$)

$\epsilon$ is how close you want the output to be to the limit

$$|f(x) - L| < \epsilon$$

Proving a limit exists

To prove that a limit exists in the mathematical sense, you need to find a $\delta$ for every $\epsilon > 0$

For every $\quad \epsilon > 0$, $\quad \delta > 0$