Unit 1: Limits &  Continuity

Limits at infinity

• useful for end behavior / horizontal asymptote

• if the limit as x goes to infinity is L, then there exists a horizontal asymptote at y = L

• Steps to find limits at infinity

  • • Find the highest power of the denominator

    • Divide every them by that power

    • Let x go towards infinity

      • • • if number/small number = infinity

          • if number/big number = 0

Infinite limits

• Have to use way 5 (looking at what happens around)

• Justification for a vertical asymptote

  • • vertical asymptote at x = a if and only if left/right hand limits goes toward infinity

Continuity

• 3 things needs to be satisfied for a function to be continuous at x = a

1. f(a) needs to be defined

2. left hand limit must equal right hand limit

3. limit = f(a)

Intermediate value theorem:

if f(x) is continuous over [a,b] and f(a) < y < f(b) or f(b) < y < f(a), then there exists at least one c in (a, b) where f(c) = y

Average Rate of Change vs Instantaneous Rate of Change

Example Problems

• Review

• Problems from Textbook