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.
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 lefthand and righthand 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.
