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# Math formulas:Useful limits

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### The General Limit Formulas

$\text{If } \lim_{x \to a} f(x) = l \text{ and } \lim_{x \to a} g(x) = m \text{ ,then}$

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

### The Common Limits

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