Calculus 1.1 Limits Graphically PDF

Calculus Write your questions and 1.1 Limits Graphically Name:_________________________ thoughts here!...

Calculus Write your questions and 1.1 Limits Graphically Name:_________________________ thoughts here! Notes What is a limit? A limit is the __________ a function ______________ from both the left and the right side of a given ___________. Example 1 y y y y y                     x x x x                                        3 3 3 3 3 lim f x   lim f x   lim f x   lim f x   lim f x   x3 x3 x3 x3 x3 Limit: (geeky math definition for Mr. Kelly) Given a function , the limit of as approaches is a real number if can be made arbitrarily close to by taking sufficiently close to (but not equal to ). If the limit exists and is a real number, then the common notation is. → What is a one‐sided limit? A one‐sided limit is the ___________ a function approaches as you approach a given ___________ from either the ______ or ______ side. Example 2 y  “The limit of as approaches 3 lim 1  from the left side is 1.” →  “The limit of as approaches 3 lim 2      from the right side is 2.” → Write your questions and 1.1 Limits Graphically thoughts here! Example 3 Notes a. lim f  x   b. lim f  x   c. lim f  x   y  x 2  x 2 x2  d. lim f x   e. lim f  x   f. lim f  x     x 1 x 0 x 3  g. lim f  x   h. lim f  x   x i. 2             x 1 x 3   j. 1    When does a limit not exist? 1. 2. 3. Example 4 y Sketch a graph of a function that  satisfies all of the following conditions.   a. 3 1  b. lim g ( x)  4  x3  x c. lim g ( x) x 2  1                 d. is increasing on 2 3  e. lim g ( x) lim g ( x) x 2  x2     Example 5 Write T (true) or F (false) under each statement. y  Use the graph on the right. a. lim f  x   1 b. lim f  x   2 c. lim f  x   1  x 1 x2 x1 d. lim f  x   2 e. lim f  x   does not exist  x1 x 1     f. lim f x   lim f  x  g. lim f x   does not exist  x0 x0 x 2 1.1 Limits Graphically Calculus Name: _____________________________ Practice For 1‐5, give the value of each statement. If the value does not exist, write “does not exist” or “undefined.” y 1.  a. lim f ( x)  x 1 b. 1 c. lim f ( x)  x 0    d. lim f ( x)  x 2  e. 1 f. 2 x          g. lim f ( x)  x 1 h. lim f ( x)  x 1 i. lim f ( x)  x 2    y 2.  a. lim f ( x)  x 3 b. 1 c. lim f ( x)  x 1    d. lim f ( x)  x 2  e. 3 f. lim f ( x)  x 2   x            g. lim f ( x)  x 2 h. 2 i. 4     y 3.  a. lim f ( x)  x 3 b. 3 c. lim f ( x)  x 0    lim f ( x)  0 lim f ( x)   d. e. f. x x 3 x 3            lim f ( x)  1 1.6  g. h. i.  x 0   y 4.  a. lim f ( x)  x 1 b. 2 c. lim f ( x)  x2    d. lim f ( x)  x 1 e. 4 f. lim f ( x)  x 1    x           g. lim f ( x)  x 1  h. 1 i. lim f ( x)  x 4      y 5.  a. lim f ( x)  x 3 b. 1 c. lim f ( x)  x 3   d. lim f ( x)  x 1 e. 3 f. lim f ( x)  x 3    x           g. 3 h. lim f ( x)  x 0 i. 4    6. Sketch a graph of a function that satisfies all of the following conditions. y a. 2 5    b. lim f ( x)  1 x 2    x lim f ( x) 3               c.    x 4   d. is increasing on 2   e. lim f ( x) lim f ( x) x 4 x 4  7. Sketch a graph of a function that satisfies all of the following conditions. 1 3 y a.   lim g ( x)  2  b.  x1   x c. lim g ( x) x3 5                  d. is increasing only on 5 3 and 1    e. lim g ( x) lim g ( x) x3 x3 8. Sketch a graph of a function that satisfies all of the y  following conditions.  a. limh( x) 2 1  x3    b. 3 is undefined. x                 c. lim lim  → →   d. is constant on 2 3 and decreasing  everywhere else. 1.1 Limits Graphically Test Prep 1. The graph of the function f is shown. Which of the following statements about f is true? (A) lim lim (B) lim 4 → → → (C) lim 4 (D) lim 1 → → (E) lim does not exist. → 2. The figure below shows the graph of a function with domain 0 4. Which of the following statements are true? I. lim exists. → II. lim exists. → III. lim exists. → (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III 3. If represents the greatest integer that is less than or equal to x, then lim → (A) 2 (B) 1 (C) 0 (D) 2 (E) the limit does not exist 4. Consider the function shown below. Which of the following statements is true? (A) lim 3 → (B) 1 1 (C) is continuous for all x. (D) lim 1 → (E) None of the above