\[ \begin{aligned} \lim_{x \to a}[f(x)+g(x)] &= \lim_{x \to a}f(x)+\lim_{x \to a}g(x)\\ \lim_{x \to a}[f(x)-g(x)] &= \lim_{x \to a}f(x)-\lim_{x \to a}g(x)\\ \lim_{x \to a}[f(x)\cdot g(x)] &= \lim_{x \to a}f(x)\cdot\lim_{x \to a}g(x)\\ \lim_{x \to a}\frac{f(x)}{g(x)} &= \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)},\ g(a)\neq0 \end{aligned} \]
\[ \begin{aligned} \lim_{x \to 0}\frac{\sin x}{x} &= 1 \\ \lim_{x \to 0}\frac{1-\cos x}{x} &= 0 \\ \lim_{x \to \infty}\left(1+\frac{1}{x}\right)^x &= e \\ \lim_{x \to 0}(1+x)^{1/x} &= e \end{aligned} \]
\[ \begin{aligned} \lim_{x \to \infty}\frac{1}{x^n} &= 0 \ (n>0)\\ \lim_{x \to \infty}\frac{x^n}{e^x} &= 0\\ \lim_{x \to \infty}\frac{e^x}{x^n} &= \infty\\ \lim_{x \to \infty}\left(1+\frac{1}{x}\right)^x &= e \end{aligned} \]