How to show \$g(x)=frac{4^x+x^2-ln(2)cdot x-1}{tan(2x)}\$ is continuous at \$x = 0\$ for \$g(0) := ln(sqrt{2})\$

\$g(x)\$ is defined on the Intervalls \$(frac{-pi}{2}, 0) cup(0,frac{pi}{2})\$ I’ve tried doing it by using L’Hôpital but \$sin(x)\$ gets into the denominator and I can’t get rid off it.

difference between \$+infty\$ and \$infty\$

I’m taking Mathematical Analysis “I” and I’m studying limits where I have limits to the infinity, but I don’t know what’s the difference between \$lim_{x to infty}\$ and \$lim_{x to +infty}\$ I …

\$lim_{xto 0}frac{sin(x)-x+frac{x^3}{3!}-frac{x^5}{5!}}{m x^n}=frac{8}{7!}\$

If \$\$lim_{xto 0}dfrac{sin(x)-x+dfrac{x^3}{3!}-dfrac{x^5}{5!}}{m x^n}=dfrac{8}{7!}\$\$ then find \$m+n\$: My attempts: note that \$\$sin(x) = x – frac{x^3}{3!} + frac{x^5}{5!} – frac{x^…

Decay of smooth functions around their boundary

If \$fin{cal C}^infty({bf R}^n)\$ and \$(x_k)_k\$ is a converging sequence with \$d(x_k,{rm supp}(f)^C)le 1/k\$, does \$k^N f(x_k)\$ converge to \$0\$ then for any fixed \$N>0\$?