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