Useful Limits Formulas

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

\( \lim_{x \to \infty}~\left(1+\frac{1}{n}\right)^n = e \)

\( \lim_{x \to \infty}~(1 + n)^{1/n} = e \)

\( \lim_{x \to 0}~\frac{\sin x}{x} = 1 \)

\( \lim_{x \to 0}~\frac{\tan x}{x} = 1 \)

\( \lim_{x \to 0}~\frac{\cos x-1}{x} = 0  \)

\( \lim_{x \to a}~\frac{x^n – a^n}{x-a} = n\,a^{n-1} \)

 

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