If f has limit l and g has limit m, prove that the limit of the maximum is the maximum of the limits
Hint: For any real $a,b\in\Bbb R,$ we have that $$\max\{a,b\}=\frac{a+b+|a-b|}2.$$ In particular, then, applying absolute value properties, we have $$\begin{align}\bigl|\max\{f(x),g(x)\}-\max\{l,m\}\bigr| &= \left|\frac{\bigl(f(x)-l\bigr)+\bigl(g(x)-m\bigr)+|f(x)-g(x)|-|l-m|}2\right|\\ &\le \frac{|f(x)-l|}2+\frac{|g(x)-m|}2+\frac12\bigl||f(x)-g(x)|-|l-m|\bigr|.\end{align}$$
Note also that $\bigl||a|-|b|\bigr|\le |a-b|$ for all $a,b\in\Bbb R$ (readily proved from triangle inequality. See if you can take it from there.