# The formulas related to Limits

#### The General Limit Formulas

 If $$\lim_{x \to a} f(x) = l$$ and $$\lim_{x \to a} g(x) = m$$ ,  then
$$\lim_{x \to a} ~\left[ f(x) \pm g(x) \right] = l \pm m$$ $$\lim_{x \to a} ~\left[ f(x) \cdot g(x) \right] = l \cdot m$$ $$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{l}{m}$$ $$\lim_{x \to a} c\cdot f(x) = c \cdot l$$ $$\lim_{x \to a} \frac{1}{f(x)} = \frac{1}{l}$$

#### The Common Limits

$$\lim_{x \to \infty}~\left(1+\frac{1}{n}\right)^n = e$$ $$\lim_{x \to \infty}~(1 + n)^{1/n} = e$$ $$\lim_{x \to 0}~\frac{\sin x}{x} = 1$$ $$\lim_{x \to 0}~\frac{\tan x}{x} = 1$$ $$\lim_{x \to 0}~\frac{\cos x-1}{x} = 0$$ $$\lim_{x \to a}~\frac{x^n - a^n}{x-a} = n\,a^{n-1}$$