Definition

Limits describe what happens around a point $c$ given a function $f (x)$ .

e.g. given a function $f (x)$ you could describe what happens around point $c$ by saying that the values of $f (x)$ get closer and closer to a value $y$ .

Limit Notation

lim_{x \to c}

lim_{x \to \pm \infty}

Rules

The limit exists if the limit from the right and left hand sides of a point both approach the same value. $$\lim_{x\to c^-}=\lim_{x\to c^+}$$
The point does not have to exist for the limit to exist. $$f(c)=\text{DNE}\quad$$however $$\quad\lim_{x\to c}f(x)=a$$

Indeterminate Forms

\infty - \infty \infty^{0} \infty * 0 1^{\infty} 0^{0} \frac{\pm \infty}{\pm \infty} \frac{0}{0}

These are forms that limits may take where the limit cannot be calculated by normal means. Usually the function has to be manipulated to find the limit.

Special Limits

Limits -> $\infty$ :

lim_{x \to \pm \infty} \frac{c}{x} = 0

Limits -> 0:

lim_{x \to 0} \frac{\sin x}{x} = 1 lim_{x \to 0} \frac{1 - \cos x}{x} = 0 lim_{x \to 0} \frac{\tan x}{x} = 1

Definition

Limit Notation

Rules

Indeterminate Forms

Special Limits

Calculating Limits

Direct Substitution

Factoring Polynomials

L'Hopitals Rule

N'th Degree Division