What are the most important well-known limits?

Question

Some of the well-known limits that I know and often use:

$$\lim_{x \to 0}{\frac{\sin{x}}{x}}=\lim_{x \to 0}{\frac{\tan{x}}{x}}=1$$ $$\lim_{x \to 0}{\frac{\arcsin{x}}{x}}=\lim_{x \to 0}{\frac{\arctan{x}}{x}}=1$$ $$\lim_{x \to 0}{\frac{e^x-1}{x}}=1$$ $$\lim_{x \to 0}{\frac{a^x-1}{x}}=\ln{a}$$ $$\lim_{x \to 0}{\frac{\ln{(x+1)}}{x}}=1$$ $$\lim_{x \to 0}{(1+x)^{\frac{1}{x}}}=e$$ $$\lim_{|x| \to \infty}{(1+\frac{1}{x})^x}=e$$

Then some less common bur perhaps also useful at times:

$$\lim_{x \to 0}{\frac{x-\sin{x}}{x^3}}=\frac{1}{6}$$ $$\lim_{x \to 0}{\frac{x-\ln{(1+x)}}{x^2}}=\frac{1}{2}$$ $$\lim_{x \to 0}{\frac{e-(1+x)^{\frac{1}{x}}}{x}}=\frac{e}{2}$$

What are some other limits that may turn out to be useful when solving the more difficult ones? Which of them are good to know?

Answer 1

$$\lim_{x \to \infty}({\frac{x}{x+k}})^{x}=e^{-k}$$ $$\lim_{x \to 0}(1+kx)^{\frac{m}{x}}=\lim_{x \to \infty}(1+\frac{k}{x})^{mx}=e^{mk}$$

$$\lim_{x \to \infty}(x)^{\frac{1}{x}}=1$$ $$\lim_{x \to \infty}{\frac{ln(x)}{x}}=0$$

You can always calculate limits using L'Hospital rule and the standard techniques. And the limits that you have mentioned as less common but useful are some of the standard questions.