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