3. Limits

continuous function + existing limit = you can pull the limit through

Important Limits

1. $\underset{n\to \infty }{\lim }\frac{\log n}{n}=0=\underset{n\to \infty }{\lim }\frac{\ln n}{n}$

---

2. $\underset{n\to \infty }{\lim }{n}^{1/n}=1$

---

3. $\underset{n\to \infty }{\lim }{x}^{1/n}=1,x>0$

---

4. $\forall x\in \mathbb{R}:\underset{n\to \infty }{\lim }{\left(1+\frac{x}{n}\right)}^n={\mathrm{e}}^{\mathrm{x}}$(including for $x=1$)

---

5. $\forall x\in \mathbb{R}:\underset{n\to \infty }{\lim }\frac{x^n}{n!}=0$

Tricks

• Take $\ln$, simplify, then raise to $\mathrm{e}$