A table of values for a function $y=f(x)$ is given below. You do not know a formula for this function.

$x$ $f(x)$ 0.01 0.174 0.02 0.213 0.03 0.241 0.04 0.265 0.05 0.286 0.06 0.304 0.07 0.320 0.08 0.336 0.09 0.350 0.10 0.363

$x$ $f(x)$ 0.11 0.375 0.12 0.387 0.13 0.398 0.14 0.408 0.15 0.418 0.16 0.428 0.17 0.437 0.18 0.446 0.19 0.454 0.20 0.463

$x$ $f(x)$ 0.21 0.471 0.22 0.478 0.23 0.486 0.24 0.493 0.25 unknown 0.26 0.507 0.27 0.513 0.28 0.520 0.29 0.526 0.30 0.532

$x$ $f(x)$ 0.31 0.538 0.32 0.544 0.33 0.550 0.34 0.555 0.35 0.561 0.36 0.566 0.37 0.571 0.38 0.576 0.39 0.581 0.40 0.586

$x$ $f(x)$ 0.41 0.591 0.42 0.595 0.43 0.600 0.44 0.604 0.45 0.609 0.46 0.613 0.47 0.617 0.48 0.622 0.49 0.626 0.50 0.630

In this question, we examine $\displaystyle{ \lim_{x \to 0.25} f(x) }$ numerically. This question asks you to predict the value of the limit. Then, it gives you several possible values of the $y$-tolerance ($\varepsilon$ value) and guides you through the procedure to find each corresponding $x$-tolerance ($\delta$ value). Finally, it asks you to reflect on how this process illustrates finding limits using the definition of a limit.