The Limit of a Function

In order to find tangents to a curve, we need to first be able to compute the limit of a function. We will start by looking at some formal definitions of a limit, and break them down so they are easier to understand.

Definition 1: The limit of f(x) as x approaches a, is written as: lim_{x \to a} f(x) = L

It essentially means, as we approach the value a, from the left and right side of the point, the value of the function gets close to L. Note that with this definition, we do not let the function actually equal to the value a, we simply approach it from either side. Let’s take a look at an example of this.

Example 1: Estimate the value of lim_{x \to 1} \frac{x-1}{x^2-1}

If we wanted to try to guess at the value of this limit, we would start by considering what the function looks like approaching from the left and right hand sides of the function. So, values to the left of 1 are values that are smaller than 1, and values to the right of 1 are values that are larger than 1. Let’s make a table to see what each of these values looks like.

You can see that in this case, both sides of the function appear to converge towards the value 0.5. So, we can conclude that as we approach the x value of 1, the function value approaches 0.5. You should note that the value at 1 is actually undefined, so we can’t know exactly what the value at 1 truly is.

It is very common to see undefined values in limits that we are solving. Often times we will need to be able to determine what value a function approaches, when there is no actual value defined. In the first example, you can note that the left and right sides of the function converged to the same value. This is a key property for limits. Let’s consider a few definitions related to this.

Definition 2: The limit of a function, approaching a from the left hand side, is written as: lim_{x \to a^-} f(x) = L

In example 1, when we were finding the values to the left of the function, we were really determining lim_{x \to 1^-} \frac{x-1}{x^2-1} = 0.5. In plain English, the limit as x approaches 1 from the left hand side is 0.5. Similarly, we can define the right hand limit of the function as well.

Definition 3: The limit of a function, approaching a from the right hand side, is written as: lim_{x \to a^+} f(x) = L

In example 1, when we were finding the values to the right of the function, we were determining lim_{x \to 1^+} \frac{x-1}{x^2-1} = 0.5

One very important feature of a limit is that the left hand side of the limit, and the right hand side of the limit must be equal to each other. This means they must both approach the same value. We can define this as follows.

Definition 4: lim_{x \to a} f(x) = L if and only if lim_{x \to a^+} f(x) = L and lim_{x \to a^-} f(x) = L.

It makes intuitive sense that if the left and right hand sides of a function do different things, we can’t make any conclusions about what is in between them. The implication of this is that some limits are not able to be solve.

Example 2: Estimate the limit lim_{x \to 0} \frac{1}{x}

In this problem, we can try to check the limit on the left of the function, and the limit on the right of the function. The limit on the left of the function approaches very large negative values. When this happens, we can say the value approaches -\infty, since it will continue to grow negatively for an infinite amount of time. On the other hand, if we approach the limit from the right hand side, we get large positive values. When this happens, we say the value approaches \infty, since the value will continue growing positively for an infinite amount of time.

As you can see, the limit on either side approaches very different values. Therefore doing a limit has told us nothing about what happens around 0. Since we can’t conclude anything about the value it approaches, we say this limit is undefined.

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