Well-Known Limits Existing Limits limx→0sinxx=1\lim_{x\to 0} \frac{\sin x}{x} = 1x→0limxsinx=1 limx→axn−anx−a=nan−1\lim_{x\to a} \frac{x^n - a^n}{x - a} = na^{n-1}x→alimx−axn−an=nan−1 limx→∞(1+ax)bx=eab\lim_{x\to \infty} \bigg(1+\frac{a}{x}\bigg)^{bx} = e^{ab}x→∞lim(1+xa)bx=eab ∀x∈R limn→∞xnn!=0\forall x\in\mathbb{R}\;\; \lim_{n\to\infty}\frac{x^n}{n!}=0∀x∈Rn→∞limn!xn=0 Limits that DNE limx→∞sinx\lim_{x\to \infty} \sin xx→∞limsinx limx→0sin(1x)\lim_{x\to 0} \sin\bigg(\frac{1}{x}\bigg)x→0limsin(x1) Indeterminate Forms 00\frac{0}{0}00 ∞∞\frac{\infty}{\infty}∞∞ ∞⋅0\infty\cdot0∞⋅0 ∞−∞\infty-\infty∞−∞ ∞0\infty^{0}∞0 000^000 1∞1^\infty1∞
