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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.

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Other differentiation topics

Acknowledgements

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