List of top Mathematics Questions on limits and derivatives

Evaluate the following limit: $ \lim_{n \to \infty} \prod_{r=3}^n \frac{r^3 - 8}{r^3 + 8} $.

If $ f(x) $ is defined as follows:
$$ f(x) = \begin{cases} 4, & \text{if } -\infty < x < -\sqrt{5}, \\ x^2 - 1, & \text{if } -\sqrt{5} \leq x \leq \sqrt{5}, \\ 4, & \text{if } \sqrt{5} \leq x < \infty. \end{cases} $$ If $ k $ is the number of points where $ f(x) $ is not differentiable, then $ k - 2 = $

Let $ f(x) = x^2 \log(\cos x) \log(1 + x) $ for $ x \neq 0 $, and $ f(0) = 0 $. Determine the behavior of $ f(x) $ at $ x = 0 $.

Let $ f $ be the function defined by:
\[ f(x) = \begin{cases} \frac{x^2 - 1}{x^2 - 2|x-1| - 1}, & \text{if } x \neq 1, \\ \frac{1}{2}, & \text{if } x = 1. \end{cases} \] The function is continuous at:

Let the function $ g: (-\infty, 0) \rightarrow \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $ be given by $ g(u) = 2 \tan^{-1}(e^u) - \frac{\pi}{2} $. Determine the properties of $ g $.