Introduction

Welcome to another lesson, and this time we will be working with limit values from tables! The concept is very similar to estimating limits from graphs. Essentially, you’re using numbers as information to solve for the limit. 

How Tables are Formed

Let’s revisit the idea of the limit. The limit is the value of the function as you get closer and closer to a certain $x$-value. If we want to have a table, it must have values that get closer and closer to it.

For example, let’s try to solve for $\underset{x\to 2}{\lim }f(x)$ using the following table below:

$\begin{array}{cccccc}x& 0& 1& 2& 3& 4\\ f(x)& 3& -2& \text{Undefined}& 4& 10\end{array}$

Notice that the $x$-value is in increments of $1$. When you are calculating a limit, you want to look at values that get closer and closer to the point. However, this table doesn’t suggest this and seems to not highlight any value. A better table to write would be this one:

$\begin{array}{cccccc}x& 1.9& 1.99& 2& 2.01& 2.1\\ f(x)& -4.8& -4.95& \text{Undefined}& -5.05& -5.2\end{array}$

Notice that the $x$ values of the graph from both sides gets closer and closer to $x=2$. This is a much better way to illustrate the limit. As we can see here, the $x$-value gets closer to $2$ from both sides, and $f(x)$ gets closer to $-5$ from both sides. We can conclude from this table that $\underset{x\to 2}{\lim }f(x)=-5$ from this graph.

Take note that some tables can also only approach from one direction. It can be used for solving one-sided limits, but not for the limit itself. 

One more thing I should note is that it is possible to figure out the left and right hand limits, and it works the same way as on a graph. You simply look at one side. For example, let’s solve for $\underset{x\to -3}{\lim }g(x)$

$\begin{array}{cccccc}x& -3.1& -3.01& -3& -2.99& -2.9\\ g(x)& -98738& -103848482& \text{Undefined}& 3.98& 3.91\end{array}$

Notice that both sides go to different values. This means that $\underset{x\to -3}{\lim }g(x)$ is undefined, but we can solve for the left and right hand limits. 

If we look at the left of $x=-3$, it seems that the value gets closer and closer to negative infinity. We can suspect that $\underset{x\to -3^{-}}{\lim }=-\infty$. If we look from the right side, the values seem to approach $4$, so we can say that $\underset{x\to -3^{+}}{\lim }=4$. 

Example 1:

The following table shows selected values of $x$ and $g(x)$. Solve for $\underset{x\to 2}{\lim }g(x)$ if it exists. If not, solve for the left and right hand limits. 

$\begin{array}{cccccc}x& 1.99& 1.999& 2& 2.001& 2.01\\ g(x)& 4.1& 4.01& 0& -199.998& -199.927\end{array}$

Firstly, we need to make sure that this table is usable. Because the $x$-values are all centered around the point we want to take the limit of, this table can be used to solve for the limit. 

Notice that $g(x)$ from the left side seems to be approaching $4$, and from the right side, $g(x)$ seems to be approaching $-200$. Because both directions go to different values, $\underset{x\to 2}{\lim }g(x)$ does not exist. However, we can solve for the left and right hand limits as we have discussed earlier.

$\underset{x\to 2^{+}}{\lim }=-200$

$\underset{x\to 2^{-}}{\lim }=4$

Example 2:

The following table shows selected values of $x$ and $h(x)$. Determine if this table can be used to solve for $\underset{x\to -6}{\lim }h(x)$ and explain why or why not.

$\begin{array}{cccccc}x& -6& -5.99& -5.9& 2& 2.01\\ h(x)& -4& 2.9999& 2.99& 80& 80.25\end{array}$

To solve for the limit, we will need the limit from both sides and see if they match up. However, notice that we only have one side, which is from the right side. This means that we don’t know if the other side matches up or not and so this table doesn’t provide us with enough information. As such, this table cannot be used to solve for $\underset{x\to -6}{\lim }h(x)$ because it only shows one side. 

Practice Problems