Limits

What is a limit?

The output (y-value) that a function is approaching as x approaches a specified value

One sided limit notation

xn-  means left side limit

xn+  means right side limit

Limit Existence Theorem

In order for a limit to exist it must have equal one sided limits

Continuity

  1. The value must exist: f(c) exists

  2. The limit must exist: xcf(x) exists

  3. They must be equal: xcf(x) = f(c)

Horizontal Asymptotes (limits)

Limits as x approaches

Vertical Asymptotes (limits)

When substituting you get an undefined value n0 

Types of Discontinuities 

  • Jump Discontinuity- one sided limits are different values

  • Removable Discontinuity- One sided limits are equal (limit exists) but not equal to the value

  • Infinite (essential) Discontinuity- vertical asymptote

xTOPbottom

Top heavy =

xtopBOTTOM

Bottom heavy = 0

xtopbottom

Equal powers = ratio of the coefficients of highest powered terms

Limit definition of derivative of f(x)

h0f(x+h)-f(x)h