Limits

This is really obvious, I was a bit lazy to write these down.

• Squeeze Theorem
• L’Hopital’s Rule

Also, in MATH119, I learned that we can use Big O to calculate these limits!

Basics

${\lim }_{n\to \infty }C=C,\text{ for any constant }C$ ${\lim }_{n\to \infty }n=\infty$ ${\lim }_{n\to \infty }\frac{1}{n}=0$

Algebra of Limits

${\lim }_{x\to a}cf(x)=c{\lim }_{x\to a}f(x)$ ${\lim }_{x\to a}[f(x)\pm g(x)]={\lim }_{x\to a}f(x)\pm {\lim }_{x\to a}g(x)$ ${\lim }_{x\to a}[f(x)\cdot g(x)]={\lim }_{x\to a}f(x)\cdot {\lim }_{x\to a}g(x)$ ${\lim }_{x\to a}[\frac{f(x)}{g(x)}]=\frac{{\lim }_{x\to a}f(x)}{{\lim }_{x\to a}g(x)},\text{ if }{\lim }_{x\to a}g(x)\ne 0$

${\lim }_{x\to a}(f(x{)}^k)=({\lim }_{x\to a}f(x){)}^k$

Sequences

A sequence an infinitely long list of numbers. Whether a sequence approaches a limit?

${\lim }_{x\to 2}C=C$ ${\lim }_{x\to 2}f(x)=5$ ${\lim }_{x\to 2}f(x)=5$