Informal Limits

Limits Informally

Below is the informal definition of a limit.

Informally, we can think of the limit as a tool that allows us to find the slope of a curve at any given point along the curve. Recall that earlier, we saw how the limit allows us to find the slope of the tangent line. The procedure of finding that slope is the method of differentiation. In later sections, we will determine how to find the area under a curve. This procedure is called the method of integration. Both methods cannot be done with algebra alone. We need limits.

Limits are what allow us to go from the secant line, an approximation of the slope at a given point, to the tangent line, a much better approximation. The diagram below is a static representation of this process. We start with some line ${\overleftrightarrow{\rm pq_7}.}$ ${p}$ remains fixed, but as we get closer and closer to ${p,}$ (going from ${q_7}$ towards ${q_1,}$) the approximation gets better and better, until we eventually get to something that looks very much like a tangent line. The end result: A secant line so close to a tangent line that the difference is negligable:

A crucial point that cannot be overstated: We never actually get to the point ${p.}$ We can get very close, but we never get there. This is because our application of the limit is restricted to the secant line. That was our starting premise. Without the secant line, the argument is nonsensical. And to have a secant line, we need two points. If we arrive at ${p,}$ we no longer have two points. Without two points, there is no secant line, and without a secant line, the argument falls apart. We never — never — get to the point ${p.}$

The idea of the limit (specifically the limit of some value ${x}$ moving closer and closer to ${0}$) represents this process of bringing some point ${q}$ closer and closer and closer to ${p.}$ Bring ${q}$ closer and closer to ${p}$ is just another way of saying that the distance between ${p}$ and ${q}$ gets smaller and smaller. Thus, that value ${x}$ that's moving closer and closer to ${0}$ is the distance between ${p}$ and ${q.}$

Limits can be investigated through algebraic manipulation; we can treat them as terms in and of themselves. This is done through the limit laws:

Axiom. Where ${a, x, C \in \reals}$:

• ${\lim\limits_{x \to a}x = a}$
• ${\lim\limits_{x \to a} C = C}$

For example, ${\lim\limits_{x \to 0}5 = 5,}$ ${\lim\limits_{x \to \pi} x = \pi.}$ We then have the following limit laws:

Limit Laws. Suppose ${f(x)}$ and ${g(x)}$ are functions defined for all ${x \neq a}$ over some open interval containing ${a,}$ where ${a \in \reals.}$ Further suppose ${L, M \in \reals,}$ where ${L = \lim\limits_{x \to a}f(x)}$ and ${M = \lim\limits_{x \to a}g(x).}$ Finally, suppose ${C}$ is a constant. The following laws hold:

1. Limit Constant Multiple Law. ${\lim\limits_{x \to a}C f(x) = C \cdot \lim\limits_{x \to a}f(x) = cL}$
2. Limit Sum Law. ${\lim\limits_{x \to a}[f(x) + g(x)] = \lim\limits_{x \to a}f(x) + \lim\limits_{x \to a}g(x) = L + M}$
3. Limit Difference Law. ${\lim\limits_{x \to a}[f(x) - g(x)] = \lim\limits_{x \to a}f(x) - \lim\limits_{x \to a}g(x) = L - M}$
4. Limit Product Law. ${\lim\limits_{x \to a}[f(x) \cdot g(x)] = \lim\limits_{x \to a} f(x) \cdot \lim\limits_{x \to a}g(x) = L \cdot M}$
5. Limit Quotient Law. ${\lim\limits_{x \to a} \dfrac{f(x)}{g(x)} = \dfrac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a}g(x)} = \dfrac{L}{M},}$ provided ${M \neq 0}$
6. Limit Power Law. ${\lim\limits_{x \to a}[f(x)]^n = \left( \lim\limits_{x \to a} f(x) \right)^n = L^n,}$ for all ${n \in \nat.}$
7. Limit Root Law. ${\lim\limits_{x \to a}\sqrt[n]{f(x)} = \sqrt[n]{\lim\limits_{x \to a} f(x)} = \sqrt[n]{L};}$ if ${n}$ is odd, for all ${L,}$ if ${n}$ is even, for all ${L \geq 0,}$ and for all ${f(x) \geq 0.}$

Limits & Continuity

In this section, we tie limits to continuity. So far, we've been using these ideas somewhat sloppily. To make best use of these concepts, we must subject them to more rigorous treatment. As a first step, let's revisit the informal definition. Suppose we have a function, ${f(x).}$ If we can get ${f(x)}$ as close as possible to some real number ${A}$ by making ${x}$ as close as possible, but not equal, to some real number ${a,}$ then we say that ${f(x)}$ has a limit of ${A}$ as ${x}$ approaches ${a.}$ We denote this proposition with the notation:

In turn, when we write ${\lim\limits_{x \to a} f(x) = A,}$ we say that ${f(x)}$ converges to ${A}$ as ${x}$ approaches ${a.}$ Some limits are relatively easy to compute. We just have to substitute:

Usually, derivatives are harder to compute than simple limits. This is because we are computing the limit of a quotient that takes functions as inputs:

This is a good example of how framing, or rewriting, propositions and expressions can dramatically change a problem. Here's an interesting point: If we suppose ${x = x_0,}$ then we always get ${\frac{0}{0}.}$ And as we know, ${\frac{0}{0}}$ is something we do not want in continuous mathematics. In other words, the assumption ${x = x_0}$ doesn't work in the context of the difference quotient. Because of this issue, we must always have some form of cancellation, or simplification, of the difference quotient. To better understand limits, we need a few more pieces of notation. The expression:

is called the right-hand limit. The right-hand limit expresses two propositions:

1. ${x}$ is approaching ${x_0,}$ and
2. ${x > x_0}$

In other words, "${x}$ is approaching ${x_0}$ from the right" (hence, the "right-hand limit"). Since ${x > x_0,}$ the two propositions allow us to infer that ${x}$ is getting smaller and smaller. In contrast, the expression:

is called the left-hand limit. The left-hand limit expresses two propositions:

1. ${x}$ is approaching ${x_0}$ and
2. ${x < x_0}$

Thus, the notation encapsulates the statement, "${x}$ is approaching ${x_0}$ from the left." Since ${x < 0,}$ this statement implies that ${x}$ is getting bigger and bigger.

The left- and right-hand limits are the component propositions of ${\lim\limits_{x \to x_0} f(x).}$ More formally:

definition. $\lim\limits*{x \to x_0} f(x) = L \nc \begin{cases} &\lim\limits*{x \to x*0^-} f(x) = L \\ \\ &\land \\ \\ &\lim\limits*{x \to x_0^+} f(x) = L \end{cases}$

Notice the direction of the implication. It only goes one way. The fact that there are left- and right-hand limits does not imply that there is a general limit. However, the fact that there is a general limit implies that there are left- and right-hand limits. As we will see with infinite discontinuities, a function can have left- and right-hand limits without having a general limit. Having these two notations allows us to decompose difficult limits. Instead of trying to tackle the limit by thinking of it all at once, we divide it into two parts, (1) the limit from the left, and (2) the limit from the right. Limits for piecewise functions are the most likely contenders for difficult computation:

Graphically, this function looks like:

What is the limit of ${f(x)?}$ The notation we just covered is helpful:

Notice that we did not use ${x = 0}$ as a value.

Introduction to Continuity

Having elaborated further on limits, we now address the concept of continuity. To begin, recall that functions come in various shapes and sizes. For example, some functions continue forever. Functions like ${y = e^x,}$ ${y = \sqrt{x},}$ ${y = \ln x,}$ and the trigonometric functions like ${\sin x}$ continue indefinitely. In contrast, there are functions that have a "hole" along their graph, like ${f(x) = \dfrac{x^2 - 1}{x - 1}.}$ Finally, there are functions with two or more branches which are not connected. E.g., ${f(x) = \dfrac{1}{x}.}$

We now formalize this phenomenon.

definition. If and only if ${f}$ is continuous at ${x_0,}$ then the limit of ${f(x)}$ as ${x}$ approaches 0 is equal to ${f(x_0).}$

In other words:

${f(x)}$ is continuous at ${x_0}$ > > ${\iff \lim\limits_{x \to x_0} f(x) = f(x_0)}$

Functions that continue indefinitely, without any holes, are said to be continuous. More specifically, we say that such functions are "continuous on its whole domain." If a function ${f}$ is not continous — i.e., does not continue indefinitely given its domain — then are three possibilities: (1) ${f}$ is not continuous at some value (a hole in the function's graph); (2) ${f}$ is discontinuous at some value (a "jump" in ${f}$'s graph'); or (3) ${f}$ has a discontinuity at some value (a "break" in the ${f}$'s graph, such as a vertical asymptote'). We will explore these possibilities in the next section.

Our definition for continuity contains numerous propositions. When we say, "${f}$ is continuous at ${x_0}$" we imply:

${\lim\limits_{x \to x_0} f(x)}$ exists.

The proposition above then implies:

${\lim\limits_{x \to x_0^+} f(x)}$ exists ${\lim\limits_{x \to x_0^-} f(x)}$ exists

I.e., that there are left- and right-hand limits for ${f(x).}$ Moreover, continuity implies that these the left- and right-hand limits are the same. Furthermore, continuity implies that ${f(x_0)}$ is defined. There can never be continuity if ${f(x_0)}$ is undefined. This is why understanding a function's domain is critical.

To summarize, to prove a function ${f}$ is continuous at ${x = a}$ we must prove that the following propositions are true:

• ${f(a)}$ is defined
• ${\lim\limits_{x \to a} f(x)}$ exists
• ${f(a) = \lim\limits_{x \to a} f(x)}$

We should be very clear about defining continuity. When we say that ${f(x_0)}$ is continuous, we are implying that:

${\lim\limits_{x \to x_0} f(x) = f(x_0)}$

The left and right-hand side of the equation above are completely independent. At no point is ${x = x_0.}$ In fact, ${x \neq x_0.}$ The limit is intended to express what happens when ${f}$ is close to ${x_0.}$ Earlier, we said that some limits are easy, in that we can just substitute values. The limit above is the opposite. These are the kinds of limits that are more difficult, in that we cannot simply just substitute values.

The expression ${f(x_0)}$ is the result of an "easy limit" — the kind of limit where we can simply substitute. The other expression, ${\lim\limits_{x \to x_0} f(x)}$ are the more difficult limits. The ones where we cannot just use substitution. Instead, ${\lim\limits_{x \to x_0} f(x)}$ is the kind of limit where we must decompose the limit into its left- and right-hand sides or find some form of cancellation, then determine whether they are equivalent to ${f(x_0.)}$

Introduction to Discontinuity

As stated earlier, not all functions are continuous; some functions are discontinuous, or discontinuous at certain points. Accordingly, we should address these functions briefly. To recap, there if a function is not continuous, there are possibilities: (1) The function ${f}$ is not continuous at some value (e.g., a hole); (2) the function ${f}$ is discontinuous at some some value (e.g., a jump); or (3) the function has a discontinuity at some value (e.g., a vertical asymptote). More formally:

For some functions, we can remove these discontinuities. For others, we cannot. Formally:

definition. Suppose ${f}$ is a function discontinuous at ${x = a.}$ If ${\lim\limits_{x \to a} f(x)}$ exists, then ${f}$ has a removable discontinuity at ${x = a.}$ Else, ${f}$ has an essential discontinuity at ${x = a.}$

Discontinuity comes in various forms. The first such form we explore are the jump discontinuities. A jump discontinuity occurs when both left- and right-hand limits exist, but are not equal.

Definition: Jump Discontinuity. A jump discontinuity exists if and only if:

1. ${\exists \lim\limits_{x \to x_0^-} f(x),}$
2. ${\exists \lim\limits_{x \to x_0^+} f(x),}$ and
3. ${\lim\limits_{x \to x_0^-} f(x) \neq \lim\limits_{x \to x_0^+} f(x)}$

The piecewise function we saw earlier is an example of a function with a jump discontinuity. One limit was 1, the other limit was 2.

Another form of discontinuity is the removable discontinuity. In a function with a removable discontinuity, the left- and right-hand limits are equal, but there is some hole, or gap, in the function's graph. For example:

Consider, for example, the following functions:

• ${g(x) = \dfrac{\sin x}{x}}$
• ${h(x) = \dfrac{1 - \cos x}{x}}$

Notice that ${g(0)}$ is undefined. It turns out, however, that ${g(x)}$ has a removable discontinuity:

Similarly, ${h(0)}$ is also undefined, and ${h}$ also has a removable discontinuity:

Based on these facts, we say that ${g(x) = \dfrac{\sin x}{x}}$ and ${h(x) = \dfrac{1 - \cos x}{x}}$ have removable discontinuities at ${x = 0.}$ The two facts above will be used extensively in later sections when we handle the derivatives of trigonometric functions.

The third form of a discontinuity is the infinite discontinuity. Consider the function ${f(x) = 1/x.}$ Its graph appears as:

Viewing this graph, ${\lim\limits_{x \to 0^+} \dfrac{1}{x} = \infty.}$ In contrast, ${\lim\limits_{x \to 0^-} \dfrac{1}{x} = - \infty.}$ The graph of ${\dfrac{1}{x}}$ demonstrates that there are, in fact, definite limits of ${\dfrac{1}{x}.}$ However, there is no such thing as ${\lim\limits_{x \to 0} \dfrac{1}{x}.}$ Notice this distinction. There are left- and right-hand limits, but there is no general limit.

Let's briefly consider the derivative of ${y = \dfrac{1}{x}.}$ We know that ${y = \dfrac{1}{x} = x^{-1},}$ so ${y' = (-1)(x)^{-2}.}$ In other words, ${y' = -\dfrac{1}{x^2}.}$ The graph:

Note how the graph looks entirely different from ${\dfrac{1}{x}}$. Newcomers to the kingdom of calculus often think that derivatives ought to look similar to their undifferentiated forms. This intuition is very tempting, and it is very wrong. More often than note, derivatives look nothing like their undifferentiated forms, and we must rid ourselves of such notions.

The graph of a derivative is a graphical representation of how the function's slope changes. In the graph of ${\dfrac{1}{x},}$ the right branch's slope gets less steep as we get closer and closer to ${x = 0.}$ This is why the right branch of ${-\dfrac{1}{x^2}}$ gets closer and closer to ${x = 0.}$ Similarly, as the right branch gets closer to ${y = 0,}$ the corresponding right branch in ${- \dfrac{1}{x^2}}$ gets closer and closer to ${y = 0.}$ The slope of ${\dfrac{1}{x},}$ however, is always negative, hence the position of the right branch of ${- \dfrac{1}{x^2}.}$ The same analysis goes for the left branch of ${\dfrac{1}{x}.}$

The graph of ${y' = - \dfrac{1}{x^2}}$ confirms our understanding of the left- and right-hand limits. ${\lim\limits_{x \to 0} y' = - \infty.}$ This general limit implies that ${\lim\limits_{x \to 0^+} y' = - \infty,}$ and ${\lim\limits_{x \to 0^-} y' = - \infty.}$

${\dfrac{1}{x}}$ and its derivative ${- \dfrac{1}{x^2}}$ is a good example of another phenomenon. The function ${y = \dfrac{1}{x}}$ is an odd function, and the function ${y' = - \dfrac{1}{x^2}}$ is an even function. This is not a coincidence. The derivative of an odd function will always return an even function.

There is one final form of discontinuities we might encounter. We colloquially call these ugly discontinuities. There are many variants of ugly discontinuities, but one that we are likely to encounter most is ${\lim\limits_{x \to 0} y = \sin \dfrac{1}{x}.}$ There are no left- or right- hand limits for this function. This appears to violate our rule regarding general limits, but there is a work-around for these functions. However, we will not worry about these limits for now. We mention them here in the event they appear.

Differentiability Implies Continuity

Now that we've had a more rigorous treatment of limits, we can now state an important theorem in calculus:

theorem. If ${f}$ is differentiable (i.e., there exists a derivative for ${f}$) at ${x_0,}$ then ${f}$ is continuous at ${x_0.}$

This theorem is what underlies the product rule and the quotient rule, two critical rules for differentiating.

Proof. The proposition to be proved: ${\lim\limits_{x \to x_0} f(x) - f(x_0) = 0.}$ We can rewrite the proposition as ${\lim\limits_{x \to x_0} \dfrac{f(x) - f(x_0)}{x - x_0} (x-x_0).}$ Taking the limit, we have ${\lim\limits_{x \to x_0} \dfrac{f(x) - f(x_0)}{x - x_0} (x-x_0) = f'(x_0) \cdot 0 = 0.}$ Thus, ${\lim\limits_{x \to x_0} f(x) - f(x_0) = 0.}$ ${\blacksquare}$