Finding Pointwise Limit Of These Functions

Question

Sequences of functions $f_n, g_n : [0, \infty) \to \Bbb R$, defined as

$f_n(x) = \dfrac{x}{1+x^n}$ and

$g_n(x) = \begin{cases} 1 & \text{if } x > \frac{1}{n} \\ nx & \text{$0\leq x\leq1/n$} \end{cases} $

Find the Pointwise Limit.

Solution Attempt

Pointwise limit for $f_n(x)$ = \begin{cases} 0 & \text{if } 0\leq x <1 \\ \frac{1}{2} & \text{if } x = 1 \\ 0 &\text{if } x > 1 \end{cases}

Pointwise limit for $g_n(x)$ = \begin{cases} 0 & \text{if } x = 0 \\ 1 & \text{if } x > 0 \\ \end{cases}

Answer 1

Your result for $g_n$ is correct.

The first comment suggests that $f_n$ is not correct. OP has the correct pointwise limit for $x \ge 1$. To fix the solution, first observe that $f_n(0) = 0$. When $0 < x < 1$, $$\frac{x}{1+x^n} = \frac{1}{\dfrac1x + x^{n-1}} \xrightarrow[n\to\infty]{} x,$$ so the pointwise limit of $(f_n)$ is \begin{cases} \color{red}{x} & \text{if } 0\leq x <1 \\ \dfrac{1}{2} & \text{if } x = 1 \\ 0 &\text{if } x > 1. \end{cases}