Example 1

Consider the limit

This limit is evaluated by letting x ≠ 0 be infinitesimal:

Let us see what happens if instead of taking x to be infinitely small we take x to be a "small" real number. We shall make a table of values of

for various small x.

x		f(x) to four places
0.1	21/9	2.3333
0.01	201/99	2.0303
0.001	2001/999	2.0030
0.0001	20001/9999	2.0003
-0.1	19/11	1.7364
-0.01	199/101	1.9703
-0.001	1999/1001	1.9970
-0.0001	19999/10001	1.9997

We see that as x gets closer and closer to zero, f(x) gets closer and closer to 2.

The table helps us to draw the graph of the curve y = f(x). Although the point (0, 2) is not on the graph, we know that when x is close to 0, f(x) is close to 2. and draw the graph accordingly. The graph is drawn in Figure 5.3.1.