limc1s1.html

Finding Limits Using Tables and Graphs

In this note we will look at a few limit problems that we cannot do using the algebraic techniques we have used so far in the course.

Example 1

Find

Solution .

We first look at what happens to the expression when x approaches 0 from the right.

In the table below, some of the x values are given in scientific notation.

x func(x) ---------------------------- .1000000000 2.593742460 0.1000000000E-1 2.704813829 0.1000000000E-2 2.716923932 0.1000000000E-3 2.718145927 0.1000000000E-4 2.718268237 0.1000000000E-5 2.718280469 0.1000000000E-6 2.718281692

Here is a table showing what happens to the expression when x gets closer to zero from the left.

x func(x) ---------------------------- -.1000000000 2.867971991 -0.1000000000E-1 2.731999026 -0.1000000000E-2 2.719642216 -0.1000000000E-3 2.718417755 -0.1000000000E-4 2.718295420 -0.1000000000E-5 2.718283188 -0.1000000000E-6 2.718281964

Based on the tables it looks like the limit is around 2.71828. The following graph supports this conclusion.

As x approaches 0, the y values on the graph are approaching a value that is a little more than 2.7. The graph is consistent with the results in the tables.

Example 2

Find

Solution .

Here is a table showing values of x approaching 2 from both sides.

x func(x) ---------------------------- 1.900000000 9.363738610 1.990000000 9.833396200 1.999000000 9.882081000 1.999900000 9.886970000 1.999990000 9.887500000 1.999999000 9.888000000 1.999999900 9.890000000 2.000000100 9.890000000 2.000001000 9.888000000 2.000010000 9.887600000 2.000100000 9.888050000 2.001000000 9.892944000 2.010000000 9.942022700 2.100000000 10.45108570

Based on the table it appears that the limit is about 9.89. Below is a graph that also confirms this.

Example 3

Find

Solution .

Below is a table showing what happens to this expression as x approaches zero from the left and the right.

x func(x) ---------------------------- -.1000000000 0 -0.1000000000E-1 0 -0.1000000000E-2 0 -0.1000000000E-3 0 -0.1000000000E-4 0 -0.1000000000E-5 0 -0.1000000000E-6 0 0.1000000000E-6 0 0.1000000000E-5 0 0.1000000000E-4 0 0.1000000000E-3 0 0.1000000000E-2 0 0.1000000000E-1 0 .1000000000 0

Note that for each value of x that was used the value of the expression came out to be zero. Based on this table alone we'd assume that as x gets closer to 0 then the expression gets closer to zero as well. However, this is not the case as shown in the next few graphs.

As we zoom in a little closer to zero we see that the y values keep oscillating between -1 and 1.

Since y does not approach a unique real value, we say that does not exist.

D. Symancyk 1/99