1. Steps to finding a limit 2. Limits approaching a value 3. One-Sided limits 4. Limits at infinity 5. Limits with radicals 6. Trig Limits 7. Limits of ln(x) and e x

1- Steps: Direct Substitution Outcomes: A number means - continuous Numerator and denominator of a rational function both = 0 - not continuous(hole) Numerator is a real #, but the denominator is 0. (approaches an vertical asymptote)

2- Limits Approaching a Value Plug in the number Example: x 2 -9 → (2) 2 -9=-5 → This equation is a continuous function

Katherine Pistorius Meghan Weisel

2- Limits Approaching a Value Plug in the number Example 2: (x 2 -4)/(x-2) → (4-4)/(2-2) =0/0 → Factor Top and Cancel → Plug in 2 → Hole (2,4)

2- Limits Approaching a Value Plug in the number Example 3: x/(x-1) → 1/(1-1) = 1/0 → Vertical asymptote at x=1 → DNE

3- One-Sided Limits Used to describe the value of the function as x approaches a specified value from a given direction. Example 1: x/(x-1) →Graph → + Infinity

4- Limits at Infinity Degree n < Degree d y=0 Degree n > Degree d No asymptote Degree n = Degree d Leading Coefficient (n/d) Example: (2x 2 -3x+1)/(5x 2 +2x-3) →leading coefficients are equal → The answer is 2/5

5- limits with Radicals The limit of an n th root is the n th root of the limit Example: → put the limit inside of the 3 rd root → answer:

6- Trig Limits Example: cos(1/x) cos(0) → the answer is: 1 → →

7- Limits of lnx and e x ln(x)= +∞ ln(x) = -∞ ln(x)=DNE e x = +∞ e x = 0