What are some neat tricks used to evaluate limits that might be otherwise a problem to deal with? I'm not asking for methods akin to L'Hopital's rule.. which is often used.
My question is geared towards garnering comparatively less known\ esoteric methods to evaluate limits.
One example is: 
On
$$x \to 0 , \cos x \to 1 \\\cos^m(ax) \sim 1-\frac{m*(ax)^2}{2} $$ My proof for this: Based on combining Taylor expansion of $\cos x$ and this $x \to 0$, so, $(1+x)^m \sim 1+mx$. Example for that formula: $$\lim_{x \to 0}\frac{\cos^3(5x)-\sqrt[3]{\cos(3x)}}{1-\sqrt{\cos x}}=\\\lim_{x \to 0}\frac{(1-\frac{3*(5x)^2}{2})-(1-\frac{\frac{1}{3}*(3x)^2}{2})}{1-(1-\frac{\frac{1}{2}*(x)^2}{2})}$$
hint: $$f(x)=e^{\ln f(x)}\\ \lim_{x \to a}f(x)^{g(x)}=e^{\lim_{x \to a}\ln(f(x)){g(x)}}$$ now not that $f(x) \to 1 \\\ln f(x) \to 0\\$ $$x \to 0 \to \ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-...\\ $$ substitute x by $f(x)-1$ because $f(x) \to 1$ so $f(x)-1 \to 1-1=0$ that mean $$\ln(f(x))=\ln({\color{Red}{(f(x)-1)}}+1)={\color{Red}{(f(x)-1)}}-\frac{{\color{Red}{(f(x)-1)}}^2}{2}+\frac{{\color{Red}{(f(x)-1)}}^3}{3}-...\sim {\color{Red}{(f(x)-1)}} $$ now look above ,and put for $\ln(f(x))$
$$ \to \lim_{x \to a}f(x)^{g(x)}=\\e^{\lim_{x \to a}\ln(f(x)){g(x)}}\\=e^{\lim_{x \to a}{\color{Red}{(f(x)-1)}}{g(x)}} $$