Limits & Continuity of Trigonometric Functions
               SUGGESTEDREFERENCEMATERIAL:
               As you work through the problems listed below, you should reference Chapter 1.6 of the rec-
               ommendedtextbook(ortheequivalent chapter in your alternative textbook/online resource)
               and your lecture notes.
               EXPECTEDSKILLS:
                  • Know where the trigonometric and inverse trigonometric functions are continuous.
                  • Be able to use lim sinx = 1 or lim 1−cosx = 0 to help find the limits of functions
                                    x→0 x            x→0     x
                    involving trigonometric expressions, when appropriate.
                  • Understand the squeeze theorem and be able to use it to compute certain limits.
               PRACTICEPROBLEMS:
               Evaluate the following limits. If a limit does not exist, write DNE, +∞, or −∞
               (whichever is most appropriate).
                 1. lim sin(2x)
                    x→π
                       4
                     1
                 2. lim (θcosθ)
                    θ→π
                     −π
                 3.  lim cscx
                        +
                    x→0
                     +∞
                 4.  lim tanx
                    x→π+
                       2
                     −∞
                 5.  lim tanx
                    x→π−
                       2
                     +∞
                 6. lim secx
                    x→π
                       4
                     √
                       2
                                                            1
                7. limsinx
                  x→0   3x
                   1
                   3
                8. limsin3x
                  x→0    3x
                   1
                9. limsinx
                  x→0   |x|
                   DNE
               10. lim1−cosx
                  x→0     4x
                   0
               11. lim cosx
                     −
                  x→0     x
                   −∞
               12. limsin2x
                  x→0    x
                   2
               13. limtan2x
                  x→0     x
                   2
               14. lim1−3cosx
                  x→0      3x
                   DNE
               15. lim arccos −x2 
                  x→∞         x2 +3x
                   π
               16. lim   3x2 2 
                  x→0  1−cos x
                   3; Video Solution http://www.youtube.com/watch?v=-heD5XuKO-g
                                                     2
                17. limtan5x
                     x→0   sin9x
                     5
                     9
                                                          2
                18. Multiple Choice: Evaluate lim tan x
                                                   x→0   x2
                     (a) −1
                     (b) 0
                      (c) 1
                     (d) −∞
                      (e) +∞
                     c
               For problems 19-23, evaluate the following limits by first making an appropriate
               substition. If the limit does not exist, write DNE, +∞, or −∞ (whichever is
               most appropriate).
                19. lim exsin(e−x)

                     x→∞
                     1; http://www.youtube.com/watch?v=BB5sQwo jIs
                20. limsin(lnx5)
                     x→1     lnx
                     5
                21.   lim esecx
                     x→π+
                       2
                     0
                22. limsin 1 
                     x→0       2
                              x
                     DNE
                             −1
                23. lim tan     (lnx)
                        +
                     x→0
                     −π
                       2
                                                            3
                      For problem 24-28, determine the value(s) of x where the given function is con-
                      tinuous.
                        24. f(x) = cscx
                               f(x) is continuous for all x 6= πk, where k is any integer.
                        25. f(x) = esinx
                               f(x) is always continuous.
                        26. f(x) =               1         on [0,2π]
                                          1−2cosx
                               f(x) is continuous for all x in [0,2π] except for x = π and x = 5π
                                                                                                              3                 3
                        27. f(x) = sin−1x
                               f(x) is continuous on its domain of [−1,1]
                                                                   π
                                          cosx if x<
                                          
                        28. f(x) =                                 4
                                                                   π
                                          
                                          sinx if x≥ 4
                               f(x) is always continuous.
                                                                                                      3sin(kx)
                                                                                                     
                        29. Find all non-zero value(s) of k so that f(x) =                                     x          if   x>0 is continuous
                                                                                                     
                              at x = 0.                                                              6k2+5x if x≤0
                               k = 1; Video Solution: http://www.youtube.com/watch?v=-Qso-2XBRZA
                                      2
                        30. Use the Intermediate Value Theorem to prove that there is at least one solution to
                              cosx = x2 in (0,1).
                                                                  2
                               Let f(x) = cos(x) − x . Since f(x) is continuous on (−∞,∞), it is also continuous
                               on [0,1]. Notice that f(0) = 1 > 0 and f(1) = cos(1)−1 < 0. Thus, the Intermediate
                               Value Theorem states that there must be some c in (0,1) such that f(c) = 0. i.e., there
                                                                                                            2                                2
                               must be at least one c in (0,1) such that cos(c) − c = 0 =⇒ cos(c) = c , as desired.
                        31. Let f(x) be a function which satisfies 5x − 6 ≤ f(x) ≤ x2 + 3x − 5 for all x ≥ 0.
                              Compute limf(x).
                                             x→1
                               −1
                                                                                          4