Limits at Infinity
We can also talk about what value f (x) approaches as x approaches infinity.
Try the following:
- The first graph shows a simple hyperbola. What is the limit when
? In other
words, what value does f (x) approach as x approaches infinity? Move the x slider so that x gets
bigger and bigger. As you will note, f (x) approaches 0 for
this example. We write 
. In this case, we can also
say that 
, because the function also
approaches 0 as you head towards negative infinity.
- Select the second example. In this example, the limit at positive
infinity is different from the limit at negative infinity.
- Select the third example. This is just a line. The limits at positive
and negative infinity do not exist, because the function's output just
keeps on getting bigger and bigger as x heads towards infinity.
You can zoom out multiple times to allow you to move the slider to bigger
and bigger values.
- Select the fourth example. This is just a sine curve. The limits at
positive and negative infinity do not exist, because the function's
output keeps oscillating up and down as bigger as x heads towards
infinity. You can zoom out multiple times to allow you to move the slider
to bigger and bigger values.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.