Limits: An Intuitive Approach SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter. of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. EXPECTED SKILLS: Given the graph of a function y = f(x), be able to determine the limit of f(x) as x approaches some finite value (as both a one-sided and two-sided limit). Know how to determine when such a limit does not exist, and if appropriate, indicate whether the behavior of the function increases or decreases without bound. PRACTICE PROBLEMS: Questions -5 refer to the function F (x), which is illustrated below.. Compute each of the following quantities. If a limit does not exist, write +,, or x x +
x F (x) (d) F () 2. Compute each of the following quantities. If a limit does not exist, write +,, or x 3 2 x 3 + x 3 F (x) (d) F (3) 2 DNE because x 3 x 3 + 3. Compute each of the following quantities. If a limit does not exist, write +,, or x x + x F (x) (d) F () 4. Compute each of the following quantities. If a limit does not exist, write +,, or x 2
x + x (d) F ( ) 2 5. Compute each of the following quantities. If a limit does not exist, write +,, or x 3 x 3 + x 3 DNE because (d) F ( 3) x 3 x 3 + Questions 6-9 refer to the graph of G(x), which is illustrated below. 3
6. Compute each of the following quantities. If a limit does not exist, write +,, or x 8 x + 8 x 8 (d) G( ) 4 7. Compute each of the following quantities. If a limit does not exist, write +,, or lim x 5 G(x) 3 x 5 + x 5 DNE because (d) G( 5) Undefined x 5 x 5 + 8. Compute each of the following quantities. If a limit does not exist, write +,, or x x + x G(x) (d) G() 4
9. Compute each of the following quantities. If a limit does not exist, write +,, or x x + 4 x G(x) DNE because (d) G() x x + Questions -2 refer to the graph of H(x), which is illustrated below.. Compute each of the following quantities. If a limit does not exist, write +,, or x 2 x 2 + + x 2 DNE 5
(d) H( 2) Undefined. Compute each of the following quantities. If a limit does not exist, write +,, or x x + x H(x) DNE (d) H() 2. Compute each of the following quantities. If a limit does not exist, write +,, or x 2 x 2 + x 2 H(x) (d) H(2) 6
2 x if x < 3. Let f(x) = 6 x 2 if < x < 3 x 6 if x 3 Sketch the graph of f(x) and use your graph to compute each of the following: lim x f(x) 2 lim x + f(x) 6 x f(x) (d) f() (e) (f) DNE because lim f(x) lim f(x) x x + Undefined lim f(x) x 3 3 lim f(x) x 3 + 3 (g) lim x 3 f(x) 3 (h) f(3) 3 7
4. Sketch the graph of a function y = f(x) which satisfies the following conditions. (There are many possible answers.) The domain is (, 2]. f() = f(2) = 5 lim f(x) = 4 x lim f(x) = x + 5. Sketch the graph of a function y = f(x) which satisfies the following conditions. (There are many possible answers.) f( x) = f(x) lim f(x) = + x + lim f(x) = 4 x lim f(x) lim f(x). x 6 x 6 + 8
6. For each of the following, determine whether the given statement is true or false. If the statement is false, give a specific counterexample. If f(x) is not defined at x = c, then lim f(x) DNE. x c { if x < False. For example, consider f(x) = if x > undefined, we have lim f(x) =. x If lim f(x) = L, then lim f(x) = L. x a x a False. For example, consider f(x) = lim x f(x) DNE because lim f(x) = 2. x + { if x < 2 if x >. Then, even though f() is. Then, lim f(x) = ; but, x 9