Here we will investigate the limit of a "messy" sine function as x approaches 0 with the aid of the software package Mathematica.

This function seems to "go crazy" as x approaches zero, but it is also "squashed" in the neighborhood of zero.  In Precalculus, we called this type of graph a "damped" function because the oscillations of the periodic sine function are confined by some other function.  Is there a limit as x approaches 0? Let us first try to evaluate the function at 0.

The evaluation failed because of division by zero at x = 0.  Thus, let's investigate numerically (using tables) whether the function is approaching a limit as x approaches 0.  [Note: the semi-colons in the commands below "suppress" the display of the "leftpoints" and "leftvalues" and just shows the leftvalues in an easy to read table.]

0.0008268795405320025602558874291092181367252`27.1668

3.57487979720165093164705006958087707906696`23.5535*^-7

4.2054779319078249129850658974094555219132`22.6661*^-8

9.3163902710972600802751665361125341`22.4089*^-9

5.45843449448699564244387270897516803018`20.8139*^-10

Both tables show that the absolute value of the y-values of the function are getting steadily smaller.  They seem to be approaching zero as x approaches zero.  Next we look at the y-values as we approach 0 from the right.

0.0008268795405320025602558874291092181367282`27.1668

3.5748797972016509316470500695808829`23.5535*^-7

4.2054779319078249129850658974094551544677`22.6661*^-8

9.3163902710972600802751665361125341`22.4089*^-9

5.45843449448699564244387270897516803018`20.8139*^-10

The right-hand values also seem to approach zero.  [The values are identical in each table because the function is "even." It has reflective symmetry through the y-axis.] It should be emphasized that the oscillating above and below the x-axis is immaterial to the fact that the absolute value of the y-values of the function steadily decrease!

The key to this problem is to note that the graph of the function oscillates between the graphs of

y = x and y = –x,

as shown in the plot below.

Thus, the function does have limit as x approaches zero.  Mathematica, the software being used here, also has a command to find limits automatically.

The analytic reasoning for this limit involves a bit of a trick.  What we need to do is separate the limit into two pieces:

We are allowed to do this because of the properties of limits.  Now note that the first piece is a "puppy," limit,x->0 [x] =0.  The second does not have a limit: limit,x->0 [Sin[1/x]] ==> Does Not Exist.  However, in limits, we DO NOT GET TO the value x is approaching.  Sin[1/x] is undefined ONLY when x = 0.  If x is merely "nearby" 0, the expression has a numerical value that is somewhere between negative and positive one (inclusive).  Thus we have the following:

limit,x->0 [x(Sin[1/x])] = limit,x->0 [x] × limit,x->0 [Sin[1/x]]
                                 = 0 × (some number)
                                 = 0

That is clever!!!