Recall the definition of a limit: we say $\lim_{x \to a} f(x) = L$ if:

For all $\epsilon > 0$ there exists a $\delta > 0$ such that $|f(x) - L| < \epsilon$ whenever $0 < |x - a| < \delta$.

Here we will look at the above function $f$, whose limit at 1 is .

Use the applet above to find how values epsilon and delta relate to each other:

When $\epsilon = 0.5$, provide a value of $\delta$ that satisfies the conclusion of the limit definition:

For the same $\epsilon$, also provide a value of $\delta$ that does not satisfy the conclusion of the limit definition:

Now make epsilon smaller. When $\epsilon = 0.25$, provide a value of $\delta$ that satisfies the conclusion of the limit definition:

For the same $\epsilon$, also provide a value of $\delta$ that does not satisfy the conclusion of the limit definition: