Formulas of Useful Limits

1) If  $\mathop {\lim }\limits_{x \to a} f(x) = l$ and $\mathop {\lim }\limits_{x \to a} g(x) = m$, then

• $\mathop {\lim }\limits_{x \to a} \left[ {f(x) \pm g(x)} \right] = l \pm m$
• $\mathop {\lim }\limits_{x \to a} f(x) \cdot g(x) = l \cdot m$
• $\mathop {\lim }\limits_{x \to a} \frac{{f(x)}}{{g(x)}} = \frac{l}{m}$, where $m \ne 0$
• $\mathop {\lim }\limits_{x \to a} c{\text{ }}f(x) = c{\text{ }}l$
• $\mathop {\lim }\limits_{x \to a} \frac{1}{{f(x)}} = \frac{1}{l}$, where $l \ne 0$

2) $\mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{1}{n}} \right)^n} = e$, where $n$ is a real number.

3) $\mathop {\lim }\limits_{n \to 0} {\left( {1 + n} \right)^{\frac{1}{n}}} = e$, where $n$ is a real number.

4) $\mathop {\lim }\limits_{x \to 0} \frac{{Sinx}}{x} = 1$, where $x$ is measured in radians.

5) $\mathop {\lim }\limits_{x \to 0} \frac{{Tanx}}{x} = 1$

6) $\mathop {\lim }\limits_{x \to 0} \frac{{Cosx – 1}}{x} = 0$

7) $\mathop {\lim }\limits_{x \to a} \frac{{{x^n} – {a^n}}}{{x – a}} = n{a^{n – 1}}$

8) $\mathop {\lim }\limits_{x \to 0} \frac{{{a^n} – 1}}{x} = \ln a$