Limits

The value that a function approaches when a variable approaches a certain value

Notation

• ${\lim }_{x\to a}f(x)=L$
• $f(x)\to L\text{ as }x\to a$
• One sided: ${\lim }_{x\to a^{\pm }}f(x)=L$

Properties

$$\underset{x\to a}{\lim }[cf(x)]=c\underset{x\to a}{\lim }f(x)$$ $$\underset{x\to a}{\lim }[f(x)\pm g(x)]=\underset{x\to a}{\lim }f(x)\pm \underset{x\to a}{\lim }g(x)$$ $$\underset{x\to a}{\lim }[f(x)g(x)]=\underset{x\to a}{\lim }f(x)\underset{x\to a}{\lim }g(x)$$ $$\underset{x\to a}{\lim }\left[\frac{f(x)}{g(x)}\right]=\frac{{\lim }_{x\to a}f(x)}{{\lim }_{x\to a}g(x)}\text{ provided }\underset{x\to a}{\lim }g(x)\ne 0$$ $$\underset{x\to a}{\lim }{\left[f(x)\right]}^n={\left[\underset{x\to a}{\lim }f(x)\right]}^n\text{ where }n\in \mathbb{R}$$

Limit of Composite Functions

$$\underset{x\to a}{\lim }f(g(x))=\underset{x\to u}{\lim }f(x),\text{ let }u=\underset{x\to a}{\lim }g(a)$$

If ${\lim }_{x\to a}g(x)$ only approaches from above or below, x → u can be treated as a one directional limit.

• above → right (+)
• below → left (-)

Solving Limits

L’Hôpital’s Rule

• Conditions
  • ${\lim }_{x\to x}\frac{f(x)}{g(x)}$
  • ${\lim }_{x\to c}f(x)=0$ and ${\lim }_{x\to c}g(x)=0$ (or infinity but we normally just do BOB0 in that case)
  • For any real number of infinity value of c
• Then
  • ${\lim }_{x\to x}\frac{f(x)}{g(x)}={\lim }_{x\to x}\frac{f\prime (x)}{g\prime (x)}$
• Can be chained many times