> [!Example] Example > Consider the sequence $(f_{n})$ given by $f_{n}:[0,\infty)\to \mathbb{R},$ $f_{n}(x)=x^{\frac{1}{n}}.$ > > For all $x\in (0,\infty),$ $\lim_{ n \to \infty }x^{1/n} = 1.$ > > As a result, the [[Pointwise Convergence of Sequence of Real Functions|limit]] of the sequence of real functions $(f_{n})$ is $f(x)=\begin{cases}0 & x=0, \\ 1 & x>0. \end{cases}$ Sketch $f_{n}$ below: ```run-python import numpy as np import matplotlib.pyplot as plt def f(n,x): return x**(1/n) step = 0.01 x = np.arange(0, 2, step) n = [1,2,3,5,20] for i in n: y = f(i, x) plt.plot(x,y,label=f'$f_{i}) plt.legend() plt.show() ``` # Properties Note that $f$ is not continuous so which shows that the limit of a sequence of continuous of functions need not be continuous. We have here also that $\lim_{ n \to \infty } \lim_{ x \to 0^+ }f_{n}(n) =0 \neq 1= \lim_{ x \to 0^+ } \lim_{ n \to \infty } f_{n}(x)$i.e. this shows the non-commutativity of limits.