In the image provided my textbook cancels out the denominator after finding the gradient of a secant without writing "undefined for values of $h = 0$". My question is that why can they cancel it out without writing that is undefined at $h = 0$. I know if you were finding the limit then you can cancel it out and the question does require finding the limit. But shouldn't the book write that it is undefined at $h = 0$ when finding the gradient and then leave it out when it is finding the limit?
Thanks
You're absolutely correct that generally if there is any ambiguity about whether or not $h = 0$ you should specify that the new expression is undefined at $h = 0$ after the cancellation. However, I think here they are justified in omitting it. Since they are asking for the secant line between $(2,8)$ and $(2+h, (2+h)^3)$, there is an underlying assumption that $h>0$ else this wouldn't be a secant line.