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A Tabular View of Limits

Let's take another look at some of the functions we have been exploring, but using a table of values in addition to the graph of the function. Each example includes a table of values of the function which approach c from the left and right.

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1. A simple line

The first graph show the line used in a previous example. Is the limit L = 0.5 when c = 1 ? In other words, does f (x) approach 0.5 as x approaches 1? Looking at the table of values, as x approaches 1 from either direction, the output value of the function approaches 0.5. Note that the table does not present a value for f (c) as this is not needed to find the limit.

2. Line with a displaced point

Select the second example. This is just like the first case, except that one point has moved. Is the limit still L = 0.5 when c = 1 ? In other words, does f (x) approach 0.5 as x approaches 1? Looking at the table of values, as x approaches 1 from either direction, the output value of the function approaches 0.5. The fact that f (c) does not equal 0.5 (it equals 1) has no effect on the limit.

3. Line with a missing point

Select the third example. This is like the previous two cases, but there is now a point missing. Is the limit still L = 0.5 when c = 1 ? In other words, does f (x) approach 0.5 as x approaches 1? Looking at the table of values, as x approaches 1 from either direction, the output value of the function approaches 0.5. The fact that f (c) is undefined has no effect on the limit.

4. Sin(x)/x

Select the fourth example. This is a more complex function, but this example is similar to the previous one with a missing point. What is the limit when c = 0 ? In other words, what value does f (x) approach as x approaches 0? As you can see from the table of values, the output value approaches 1 from both directions.

5. A jump discontinuity

Select the fifth example, a jump discontinuity. What is the limit when c = 1? In other words, what value does f (x) approach as x approaches 1? Note that in this case, the table shows different values for the left-hand and right-hand limits. Hence there is no general limit at c = 1.

6. A vertical asymptote

Select the sixth example, a function with a vertical asymptote. What is the limit when c = 1? In other words, what value does f (x) approach as x approaches 1? The table shows that the output value gets bigger and bigger as you approach 1 from either direction, hence there is no limit.

7. A wiggly function: sin(1/x)

Select the seventh example, the wiggly sin(1/x). The table shows that the value jumps around as you approach 0, hence the limit does not exist there.

Other differentiation topics

Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.