Finding Pointwise Limit Of These Functions 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}

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}
