Introduction to Limits

Conventionally, functions will return an output when given an input. Take $f(x) = x^2\,,$ the most basic parabola. When you substitute $x$ with any number, the y-value can be calculated in a straightforward fashion. For example, when $x = 2\,,$ $f(x) = 2^2 = 4\,.$

When we introduce rational functions, division by zero complicates the process. Fig. 1 shows the graph of $f(x) = \frac{(x+5)(x-2)}{(x-2)}\,.$

When we try to compute with $x = 2\,,$ $f(x) = \frac{7 \cdot 0}{0} = \frac{0}{0}\,.$ This is undefined, and is shown by the hole in Fig. 1. However, when we get closer and closer to $x = 2\,,$ the $(x-2)$ factors are both nonzero and cancel out, so the y-value approaches $2 + 5 = 7\,.$

Limits enable the representation of y-values that may not exist in the function, but are approached from both sides. This is extremely useful for the questions calculus answers, such as “What is the slope of the tangent line at any point on this function?” and “What is the area under a curve?”

The limit has a conventional notation, and we can represent Fig. 1’s output at $x = 2$ with $\lim\limits_{x \to 2} f(x) = 7\,.$

This reads as “The limit of f(x) as x approaches 2 is equal to 7”. The $\lim$ symbol means that a limit is being taken, the $x \to 2$ below it means we take the limit of the function as x approaches 2, and the $f(x)$ represents the function which we are taking the limit of.

When taking a limit, the intuitive way to think about it is “what pattern is observed as we get infinitely close from both sides?”

Fig. 2 shows the graph of $g(x) = \frac{(x-3)^2 (x+1) (x-1)}{x-1}\,.$

To find $\lim\limits_{x \to 1} g(x)$, you can refer to a table of input/output values. The following table shows what they look like as $x$ approaches 1 from the left, which can be notated as $\lim\limits_{x \to 1^-} g(x)$ and read as “the left hand limit of g(x) as x approaches 1”.

This table illustrates outputs as $x$ approaches 1 from the right, notated as $\lim\limits_{x \to 1^+} g(x)$ and read as “the right hand limit of g(x) as x approaches 1”.

With limits, x is evaluated infinitely close to the value it approaches. In this example, the limit is obviously equal to eight, as shown by both the left and right tables.

The phrase “infinitely close” is hand-waving how a limit is actually defined, and the formal definition will be revealed later in this section.

For a limit to exist, its left-hand and right-hand limits must be equal. Let’s take a look at a piecewise function:

Fig. 3 shows the graph of this piecewise function, which appears as two disjoint lines.

Let’s use our table approach and try to find the limit on each side. The following table represents $\lim\limits_{x \to 2^-} h(x)\,.$

From the left-hand side, the limit approaches 6. Let’s try $\lim\limits_{x \to 2^+} h(x)\,.$

From the right-hand side, the limit approaches 3. As demonstrated by the tables, both the left-hand and right-hand limits exist, but since they are unequal, the limit $\lim\limits_{x \to 2} h(x)$ does not exist.

To generalize, a limit only exists if the left-hand and right-hand limits are equivalent.

In this chapter, we used tables and manually calculated output values as input values approached the value the limit was taken at. This isn’t a very meaningful way to use limits, and the table method was employed for illustrative purposes. As this series continues, the utility of limits is found within the concept of approaching a value and the clever ways one can manipulate them.

That is all for the introduction of limits, and until next time, I am out.