1.2   One-Sided Limits

EXERCISE SOLUTIONS

Problems #1-4 refer to the function,

                                                

 

with graph

 

1.            2.           3.           4.  

 

Problems #5-8 refer to the function,  

5.  

6.  

7.  

8.    

 

9.   Suppose that g is an even function, and that 

      One may deduce that  

 

The symmetry of even function graphs demands that this also equals 3.   For questions like this one, it helps greatly to draw some sketches.  The following even function was selected to help clarify problems #9 and 10:

You can see that x approaching 1 from the left will yield a graph height of 3, the same as you get with x approaching –1 from the right, due to the graph’s bilateral symmetry. 

10.   Given the information in the previous problem, you may also conclude that

We can conclude nothing, since the function may have discontinuity points like those depicted in the graph.   To see why, inspect the above graph. 

11.   Redo #9, but now assume g is an odd function. 

The symmetry about the origin of odd function graphs demands that this limit equal –3.   For questions like this one, it helps greatly to draw some sketches.  The following odd function was selected to help clarify problems #11 and 12:

 

12.   Redo #10, but now assume g is an odd function. 

We can conclude nothing, since the function may have discontinuity points like those depicted in the graph.   To see why, inspect the above graph.