From this table we can see that gets as close as one wishes to for some values that are sufficiently close to . But this does does not satisfy the definition of the limit, at least, not yet.
Indeed, let’s consider a couple more values (again, round to 3 decimal digits); What do these two tables tell us about the limit of as goes to ?
As another example of how data can be misleading, recall the following graph:
When looking at a graph, it can be easy to see what is happening, because we have all the information except the one single point we want to know about. In the graph above it should be pretty clear what the value of should be at , because we have all the values in the area around, including the values arbitrarily close to . But the solid line of the graph represents an infinitely large number of data points. Clearly there aren’t many (or any!) experiments where we get an infinitely large dataset. So in reality we are much more likely to get a graph that looks something like this:
The picture above is just a scattering of dots, which makes it difficult to get a sure sense of the pattern. “But surely it’s obvious that it is coming to a maximum at ” you might say, after all we’ve seen the original curve. But herein lies the problem; there are a lot of possible curves that could go through those dots! Consider the following graph:
This graph hits all the given data points, but clearly does something completely unexpected at the area. Why didn’t we detect this striking change in behavior? Because we didn’t have any data points whose values were “suitably close” zero! In fact, ensuring that you have data “close enough” to the -value of interest is actually almost always the hard part about limits!
In the video below we explore this issue of being “close enough” in more detail.
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Optional Content
If you are interested in how mathematicians actually prove that values are “close enough” in general (not just for limits!) you can watch the following video on how mathematicians tackle the problem of being arbitrarily close rigorously. [URL to YouTube video, but not an embedded video.]