Limits

Revisiting an Old Problem (Watch Video)

Limits are going to help us overcome a problem that we once ignored in our previous math courses, and it will also help us avoid a couple of future issues. Let’s begin by revisiting the following function: $$ \begin{aligned}
f(x) &= \frac{x^{2} – 1}{x – 1} && \text{Let } x = 1 \\
f(1) &= \frac{(1)^{2} – 1}{(1) – 1} && \text{Simplify} \\
&= \frac{0}{0} &&
\end{aligned}$$ Previously we were taught that we cannot divide by \( 0 \), and so the function is said to be undefined at \( x = 1 \). But, what if we wanted to know what y-value the function would have hit if \( x \) could be \( 1 \).

We discover that there are 3 primary approaches to answering this question: Numerical, Graphical, and Analytical.

The Numerical Approach (Watch Video)

The numerical approach involves choosing values close to the undefined x-value. In the above example, we might choose numbers closer and closer to \( x = 1 \). A convenient table might look like: $$ \begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & 0.9 & 0.99 & 0.999 & 1 & 1.001 & 1.01 & 1.1 \\
\hline
f(x) & 1.9 & 1.99 & 1.999 & ? & 2.001 & 2.01 & 2.1 \\
\hline
\end{array} $$ If we substitute these values into the function, we can see that as \( x \) approaches \( 1 \) from the left and from the right, then the y-values begin to approach \( 2 \). Thus, we can conclude that the function would have hit a y-value of \( 2 \) if \( x \) was allowed to be \( 1 \). This can be said with a convenient notation: $$ \lim_{x \rightarrow 1} \frac{x^{2} – 1}{x – 1} = 2 \ $$ This is read as “The limit as \( x \) approaches \( 1 \) of function is \( 2 \)”. Note that \( f(1) \) was undefined for this function, so: $$ \lim_{x \rightarrow 1} \frac{x^{2} – 1}{x – 1} \neq f(1) \ $$ In other words, a limit describes what a function looks like it might do as we get closer and closer to a particular x-value, but that might be different from what the fuction actually does at that x-value.

The Graphical Approach (Watch Video)

The graphical approach really only involves taking a few points borrowed from the numerical approach, and creating a graph. This will allow us to see what y-value it appears the function is approaching. So, let’s take our table of points from our continued example and graph them.

A small empty circle, referred to as a “hole”, corresponds with our undefined value at \( x = 1 \). As we approach the hole from both the left and right sides, the y-value does appear to approach \( 2 \).

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