You could evaluate this limit algebraically.
Recall that: $x^3 − 8 = (x − 2)(x^2 + 2x − 4)$
Or you could recognize that this is Ana's Definition of the Derivative.
$f(x) = x^3$
$a = 2$
So the limit $= f^\prime(2)$

You should recognize that this is Hanah's Definition of the Derivative.

Deconstruct Hanah's definition.
1. Determine what is $f(x)$.
2. Find its derivative. That is the value of the limit.

This is Hana's Definition of the Derivative, evaluated at $x=\frac{\pi}{4}.$

This is Ana's Definition of the Derivative, evaluated at
$x = π$.

This limit could be rewritten as $\lim\limits_{x\rightarrow \pi}\frac{\text{tan}(5x)-\text{tan}(5\pi)}{x-\pi}$'.
Of course, $\operatorname{tan}(5,π) = 0$.