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 $f$, which has no limit at 1. Therefore the claim that $\lim_{x \to 1} f(x) = 2$ is false, meaning the above definition fails.

First show that for some values of $\epsilon$, there are values of $\delta$ satisfying the conclusion of the limit definition. For example:

If $\epsilon = 0.7$ then provide a value for $\delta$ making the conclusion of the limit definition true:

However, not every value of $\epsilon$ will work. Use the applet to find a value for $\epsilon$ such that no $\delta$ will satisfy the conclusion of the limit definition, and record that $\epsilon$ here: