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