University:
South Hills High School
Course:
AP Calculus AB
Academic year:

2024
Views:

62

Pages:

5
Author:

M. Perez

CH.1 LIMITS ** When evaluating limits, we are checking around the point that we are approaching, NOT **Every time we find a limit, we need to check from the left and the right hand side (Only if there is a BREAK at that point). lim/(x)= S normal limit (approach a from both sides) (limf(x)= Find the limit of f(x) as x approaches a) *"If left and right hand limits DISAGREE, then the limit Does Not Exist (DNE) at that point. **If left and right hand limits AGREE, then the limit exists at that point as that value. lim f(x)= right hand limit (approach a from the right only) lim f(x)= left hand limit (approach a from the left only) Breaking Points My term "Breaking points" is a made-up term for the points where a graph is "broken(disconnected)." **Breaking Points are points on the graph that are undefined or where the graph is split into pieces. Breaking Points: 1) Holes (when the numerator and denominator equals 0) (REMOVABLE DISCONTINUITY) 2) Asymptotes (when the denominator equals (0) 3) Piece-wise functions (the where the graph is split) Note: In general when doing limits, 18 =0 ** In limits, if the two sides of a graph don't agree, then the limit does not exist. CONTINUITY Continuous functions have no breaks in them. Discontinuous functions have breaks in them (Asymptotes or Holes/Open Circles). **To check for continuity at "a", you must check left hand limits lim f(x), right hand limits lim f(x) and the value of the function at that point f(a). If all three are equal then the function is continuous at a. If f(a) lim f(x)= lim f(x) then the function is continuous at a If f(a) is not equal to either one-sided limit, then the function is not continuous (discontinuous) at 1-5 Infinite Limits ASYMPTOTES (%) (Since the point DNE we have to check a point that is close on the side we we app There are three possible answers when checking near the breaking point (the #that makes bottom 1) 00 → If we get a positive answer the limit approaches 2) - If we get a negative answer the limit approaches- 3) DNE → If we get a positive answer on one side and a negative answer on the other side, then the limit DNE EX#1: lim 1 EX#2: lim 1 1-3 = EX#3: lim 1 -3DNE EX#4: lim 2 (x-3) EX#5: lim -6 (x-8) EX#6: lim 4 -5x+5 EX#7: lim COS.t EX#8: lim ONE EX#9: lim tanx= 3-5 Limits at Infinity LIMITS THAT APPROACH INFINITY Check the powers of the numerator and denominator. 1) If the denominator (bottom) is a bigger power the limit= 0. 2) If the numerator (top) is a bigger power the limit or 3) If powers are the same the limit = coefficient of the highest power of numerator coefficient of the highest power of denominator EX#1: lim 2+5 2 = 3-7 3 EX#2: lim 8x +1 0 nack EX#3) lim 2-1 EX#4: lim 4r 8-11x EX#5: lim 7x+2 8-10 EX#6: lim 4-7-204 37+2 3 they 0 4-7-20 01-= EX#7: lim -37+2 FINDING VERTICAL ASYMPTOTES AND HOLES A vertical asymptote is the # that makes only the denominator =0. A hole occurs at the points that make the numerator and denominator = 0 at the same time. EX#1: