Given the following step function: $$ f_n(x) = \left\{ \begin{array}{ll} 0 & \quad 0 \leq x < 1 - \frac{1}{n} \\ n & \quad 1 - \frac{1}{n} \leq x < 1 \\ 0 & \quad x = 1 \\ \end{array} \right. $$ Find the following limit as x ∈ $ [0,1] $: $\lim_{n \to \infty} f_n(x)$
It is noteworthy that while the interval for x is given here for this step function, n is the variable here. How do we calculate this limit? Do we evaluate the limits at three different intervals for x?