Limits-Trigonometric Functions 
                                                                                                              
             When calculating trig limits remember to consider the following:  
                     
                  1. Remember to still plug in the x- value first.  If you get a numerical answer, that is your limit. 
                  2. If you get  0  after plugging in the  x-value, that means there is a hole, and like the other  problems with 
                                         0
                        holes, there is a limit.  Because we cannot factor, cancel and plug the x-value into what the function left 
                        after factoring, we will look at several of these situations to see if there is a pattern to these limits that we 
                        can use. 
                  3. If you get  1   or   #  , then there is a vertical asymptote at that x-value  and the limit DNE.  All of the basic 
                                         0         0
                        trig have asymptotes that go in opposite directions around the vertical asymptote. We will not pursue 
                        which direction the graph is heading with trig limits unless the function is squared or absolute valued. 
                   
              
             1.  lim sinx=   _____             2.                    lim      sin x =_____                  3.   lim cosx=   _____               4. lim                     cosx=_____ 
                  x→5π                                             x→−3π                                       x→2π                                              x→11π
                        6                                                  4                                         3                                                 6
              
              
              
              
             5.  lim sin6x=   _____                             6.  lim (sin2x−cos6x)=_____                                     7.  lim 7cosx=_____             
                  x→2π                                               x→5π                                                           x→5π
                       3                                                  6                                                               4
              
              
              
              
              
             8.  lim cscx=  _____                               9. lim cscx=_____                                  10.   lim secx =_____        11.  lim secx =_____ 
                  x→−π                                              x→4π                                                  x→π                                        x→7π
                        4                                                 3                                                    3                                           6
              
              
              
              
              
             12. a.  lim tan x=   _____                         13.  lim tan x =_____                              14. a.  lim cot x=   _____      15.   lim cot x=_____ 
                       x→π                                            x→5π                                                   x→5π                                        x→5π
                            3                                               4                                                      6                                           3
              
              
              
              
             Let’s take a look at the following limit: 
              
             Example 1  lim sinx = 0  
                                  x→0        x         0
             If 0 is plugged into the x of the function.  We know that this means that there 
             is a hole in the graph!  Let’s take a look at the picture and see if we can 
             determine the y-value of the hole, which is the limit that we are looking for.  
              
                                                                                                     sin x                                     
             It appears that the y-values are approaching 1.  lim                                             =1
                                                                                              x→0       x
                                                                                                                    
         Now examine the graphs of the following functions and give their limits:
          
         Example 2                         Example 3                          Example 4                          Example 5 
          
          lim sin4x =______            lim sin4x =____                            lim sin6x =____                   lim sin x =____ 
          x→0    x                        x→0 2x                              x→0 2x                            x→0 2x
                                  
                                  
                                  
                                  
                                  
                                  
          
         What do you notice about the coefficients of the x’s and the answer to the limit?  lim sinax =                   
                                                                                                       x→0 bx
          
         Does this Pattern work for cos x in a similar pattern? 
          lim cos3x = 1  This means that there is a vertical asymptote on this graph at x=0.  Always remember to plug in 
          x→0 6x        0
         the x-value first!   
          
         Because tanx = sinx   the pattern also works for tanx.   lim tan5x =                 
                            cosx                                          x→0 3x
          
          
          
         There is one other famous trig function that does not show up a lot, but here it is: 
          lim1−cosx = 0  , which means there is a hole and means that there is a limit.  We 
          x→0     x       0
         could use something called the squeeze theorem, but we will just look at the graph 
         and determine the answer.  
          
         After looking at the graph,  lim1−cosx = 0 or lim cosx−1 =0 . 
                                         x→0     x               x→0     x
          
          
        
       Mixed Problems:  Remember- The first thing you always do is……Plug in the number!   
                                                                   x2 +1
       1.     lim(2sinx+3cosx)                         2.     lim         
               x→0                                            x→01−sinx
        
        
        
        
        
                    x                                               x
       3.     lim                                      4.     lim        
               x→0 tan x                                      x→0 sin3x
        
        
        
                                                                               Split this one up first 
                                                                               and take the limit of 
                                                                               each piece and add 
                                                                               together. 
       5.     lim tan2x                                6.     lim x+sinx       
               x→0  x                                         x→0    x
        
        
                       Split this one up first 
                       and take the limit of 
                       each piece and 
                       multiply together. 
        
                  tan2 x
              lim       =
       7.      x→0 2x                                                8.     lim tan5x  
               tanx⋅tanx   tan x tan x                                      x→0 tan2x
                         =      ⋅
                  2x        2x     1
        
        
        
        
        
        
       9.     lim3xsecx                                              10.    limcosx−1                   
               x→0                                                          x→0   x
               
        
        
                                                                                    Be Careful- Remember the Golden Rule 
                                                                                    of Limits! 
       11.    limsin3x+tan5x                                         12. lim tan2x  
               x→0      x                                                x→0  3