Limiting Function for uniform convergence

Question

$\lim_{n\to\infty} \frac{x+n}{n}$ I assumed that this limit equaled 1, correct? I am trying to determine a limiting function.

Answer 1

As Kavi points out the limit is $1$. We can write $\frac{x+n}{n} = \frac{x }{n}+ \frac{n}{n} = \frac{x }{n}+1 = 1+ \frac{x }{n}$. The second term tends to 1. So the functions $f_n:\mathbb R \to \mathbb R; f_n(x) =\frac{x+n}{n} $ converge to $f(x)=1$ pointwise.

However the convergence is not uniform since uniform convergence means $\max_x |f_n(x)-f(x)| \to 0$ but in this case $\max_x |f_n(x)-f(x)| = \max_x |x|/n = \infty$.