1. ## Pointwise limits

I'm trying to understand how to do pointwise limits
a)f_n(x)=nx/(1+nx^2)
b)g_n(x)=(nx+sin(nx))/2n
lim[1/2+sin(nx)/2n)]
c)h_n(x)=x/(1+x^n)
d)f_n(x)=1 if |x|>=1/n
=n|x| if |x|<1/n

2. When you study pointwise limits, $x$ is a constant.

For example

$\displaystyle\lim_{n\to\infty}f_n(x)=\lim_{n\to\in fty}\frac{nx}{1+nx^2}=\lim_{n\to\infty}\frac{x}{\f rac 1n+x^2}=\frac{x}{x^2}=\frac 1x.$

Also, you can conclude the convergence isn't uniform on $\mathbb R$ 'cause each $f_n$ is continuous but the limit is not continuous.