cheat sheet for limits

Important Standard Limits

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

\(2). \displaystyle \lim_{x \to 0} \frac{\tan x}{x} = 1 \)

\(3). \displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \)

\(4). \displaystyle \lim_{x \to 0} \frac{\sin ax}{bx} = \frac{a}{b} \quad (a,b \text{ constants}) \)

\(5). \displaystyle \lim_{x \to 0} (1+x)^{1/x} = e \)

\(6). \displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e \)

\(7). \displaystyle \lim_{x \to 0} \frac{e^x - 1}{x} = 1 \)

\(8). \displaystyle \lim_{x \to 0} \frac{\log(1+x)}{x} = 1 \)

\(9). \displaystyle \lim_{x \to 0^+} x^k \log x = 0 \quad (k>0) \)

\(10). \displaystyle \lim_{x \to 0^+} x^a = 0 \quad (a>0) \)

\(11). \displaystyle \lim_{x \to \infty} \frac{\log x}{x} = 0 \)

\(12). \displaystyle \lim_{x \to \infty} \frac{x^k}{e^x} = 0 \quad (k>0) \)

\(13). \displaystyle \lim_{x \to 0} (1+x)^n = 1 \quad (n\in\mathbb{R}) \)

\(14). \displaystyle \lim_{x \to 0} \frac{a^x - 1}{x} = \ln a \quad (a>0) \)

\(16). \displaystyle \lim_{x \to \infty} \left(1+\frac{k}{x}\right)^x = e^k \quad (k\in\mathbb{R}) \)

\(17). \displaystyle \lim_{x \to 0} \frac{\arcsin x}{x} = 1 \)

\(18). \displaystyle \lim_{x \to 0} \frac{\arctan x}{x} = 1 \)

\(19). \displaystyle \lim_{x \to 0} \frac{\sinh x}{x} = 1 \)

\(20). \displaystyle \lim_{x \to 0} \frac{\tanh x}{x} = 1 \)

\(21). \displaystyle \lim_{x \to 0} \frac{\sin x}{\tan x} = 1 \)

\(22). \displaystyle \lim_{x \to \infty} \frac{\sqrt{x^2 + a}}{x} = 1 \quad (a>0) \)

\(23). \displaystyle \lim_{x \to 0^+} x^x = 1 \)

\(24). \displaystyle \lim_{x \to 0} \left( \cos x \right)^{1/x^2} = e^{-1/2} \)

\(25). \displaystyle \lim_{x \to 0} \left( \frac{\sin x}{x} \right)^{1/x^2} = e^{-1/6} \)

\(26). \displaystyle \lim_{x \to 0} \frac{1-\cos(ax)}{x^2} = \frac{a^2}{2} \quad (a\in\mathbb{R}) \)

\(27). \displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \)