List of basic formulas and rules that can help in calculating limits:
- Limit of a constant:
[
\lim_{x \to a} c = c
]
(where (c) is a constant) - Sum limit:
[
\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)
] - Difference limit:
[
\lim_{x \to a} (f(x) – g(x)) = \lim_{x \to a} f(x) – \lim_{x \to a} g(x)
] - Product limit:
[
\lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x )
] - Limit of the quotient:
[
\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad g(a) \neq 0
] - Degree limit:
[
\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n
] - L’Hôpital’s rule (for uncertainties of the form ( \frac{0}{0} ) or ( \frac{\infty}{\infty} )):
[
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}
] - Limits of trigonometric functions:
[
\lim_{x \to 0} \frac{\sin x}{x} = 1
]
[
\lim_{x \to 0} \frac{1 – \cos x}{x^2} = \frac{1}{2}
] - Limits of exponential and logarithmic functions:
[
\lim_{x \to \infty} (1 + \frac{1}{x})^x = e
]
[
\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1
]