I wonder if there's a way to double-check your answer in delta epsilon proofs, in other words if the chosen delta satisfy the conditions.
For example:
- Proof the limit of the following- $$ \lim_{x \to 1} f(x) = 1 \quad \text{for} \quad $$ $$ \begin{equation*} f(x) = \begin{cases} x^2 \quad x\neq1 \\ 2 \quad x=1 \end{cases} \end{equation*} $$
I choose delta as $$ \delta = 1- \sqrt{1-\epsilon} $$
Is there a way I can verify such a case?
edit: adding my solution:
$$ |x^2-1| < \epsilon \\ -\epsilon < x^2 -1 < \epsilon \\ \sqrt{1-\epsilon}<x< \sqrt{1+\epsilon} \\ \text{ Im looking for delta in the interval:} \quad (\sqrt{1-\epsilon},\sqrt{1+\epsilon})\\ \delta = x_0 - \sqrt{1-\epsilon}\\ \delta = 1 - \sqrt{1-\epsilon}\\ $$