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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