Using tables to approximate limit values

Limits are a tool for reasoning about function behavior, and tables are a tool for reasoning about limits. One nice thing about tables is that we can get more precise estimates of limits than we'd get by eyeballing graphs.

When using a table to approximate limits, it's important to create it in a way that simulates the feeling of getting "infinitely close" to some desired $x$‍-value.

Imagine we're asked to approximate this limit:

Note: The function is actually undefined at $x=2$‍ because the denominator evaluates to zero, but the limit as $x$‍ approaches $2$‍ still exists.

Step 1: We'd like to pick a value that's a little bit less than $x=2$‍ (that is, a value that's "to the left" of $2$‍ when thinking about the standard $x$‍-axis), so maybe start with something like $x=1.9$‍.

Step 2: Try a couple more $x$‍-values to simulate the feeling of getting infinitely close to $x=2$‍ from the left.

Notice how our $x$‍-values $\{1.9,1.99,1.9999\}$‍ really "zoom in" around $x=2$‍. A worse choice of $x$‍-values would have been constant increments like $\{-1,0,1\}$‍, which aren't very helpful for thinking about getting infinitely close to $x=2$‍.

Step 3: Approach $x=2$‍ from the right just like we did from the left. We want to do this in a way that simulates the feeling of getting infinitely close to $x=2$‍.

(Note: We've removed $x=2$‍ from the table to save space, and also because it isn't necessary for reasoning about the limit value.)

Looking at the table we've created, we have very strong evidence that the limit is $0.25$‍. But, if we're honest with ourselves, we must admit that what we have is only a reasonable approximation. We can't say for sure that this is the actual value of the limit.

Here are a several things to watch out for as you create your own tables to approximate limits:

Assuming the function value is the limit value: The example above highlights a case where the function is undefined, yet the limit still exists. Avoid jumping to conclusions about the limit value based on the function value.

Not getting infinitely close: Getting infinitely close means we’re trying to get so close to a desired $x$‍-value that there’s very little room left between where we are and where that value is—close enough to convince us that the estimate we’re getting is most likely what the limit is.

Not approaching from both sides: Remember to approach your desired $x$‍-value from both the left and the right. Remember, for the limit to exist, the left- and right-hand limits must be equal. Avoid jumping to conclusions about the limit value after only approaching your desired $x$‍-value from one side.

Assuming "left side" means "negative": Some students mistakenly believe that when approaching from the left they must use negative numbers. In the example above, we approached $x=2$‍ from the left by using positive values that were just a little bit less than $2$‍, such as $1.9$‍ and $1.99$‍. Don't assume you must use negative $x$‍-values when approaching from the left.

Confusing the limit value with the function value: Remember that the limit of a function at a certain point isn't necessarily equal to the function's value at that point. For example, in Problem 2, $g(5)=6.37$‍ but ${\displaystyle \underset{x\to 5}{lim}g(x)}$‍ is about $3.68$‍.

Thinking a limit value is always an integer: Some limits are "nice" and have integer values or nice fraction values. For example, the limit in our first example here was $0.25$‍. Some limits are less nice, like the limit in Problem 2 which is somewhere around $3.68$‍.