1.1: Defining a Limit

We base all topics in calculus on one important topic—the limit. Limits are a fundamental tool that enables us to describe how a function behaves as we change its input. In this section we define limits and present numerical and graphical methods to investigate them. We discuss the following topics:

• Definition of a Limit (Intuitive)
• One-Sided Limits
• Limits with Infinity

Definition of a Limit (Intuitive)

A limit is the number that a function \(f(x)\) approaches as its input \(x\) approaches some number \(a.\) If \(f(x)\) gets closer and closer to \(L\) as \(x\) gets closer and closer to \(a\) (without necessarily being \(a\)), then we say \[\lim_{x \to a} f(x) = L \pd\] This notation is read as the following: The limit of \(f(x),\) as \(x\) approaches \(a,\) is \(L.\) We could also use arrow notation by writing \(f(x) \to L\) as \(x \to a,\) meaning \(f(x)\) tends to \(L\) as \(x\) is made close to \(a\) (but \(x \ne a\)). In other words, we can make the values of \(f(x)\) arbitrarily close to \(L\)—as close to \(L\) as we please—by making \(x\) sufficiently close to \(a\) but not equal to \(a.\) We will present a more precise definition in Section 1.5, in which we study the interpretation of the phrase as close to \(L\) as we please.

As an example, the following table shows values of \(f(x) = x^2\) for selected values of \(x.\)

Pay attention to the statement but not equal to \(a.\) When we find limits, we have no interest in \(f(a)\) itself. In fact, \(f\) doesn't need to be defined at \(a.\) Instead, we only concern ourselves with the behavior of \(f\) near \(a.\) A limit only tells us what number \(f\) approaches as \(x \to a.\) For example, in Figure 2A \(f(a)\) is defined but in Figure 2B \(f(a)\) is undefined. Yet in both cases, \(f(x)\) approaches \(L\) as \(x\) approaches \(a\)—meaning both graphs satisfy \(\lim_{x \to a} f(x)\) \(= L.\)

It's important to realize the limitations of using graphs and tables to estimate limits. Often, it's difficult to know how close to make \(x\) to \(a,\) and technology can mislead us. For example, in Example 2 if you make \(x\) small enough, then your calculator may return \(f(x) = 0.\) This result arises because for very small values of \(x,\) your calculator cannot hold enough digits to distinguish between \(\sqrt{4 + x}\) and \(2.\) Thus, the calculator returns the difference \(\par{\sqrt{4 + x} - 2}\) as \(0.\) But the limit \(\lim_{x \to 0} \par{\sqrt{4 + x} - 2}/x\) turns out to be \(1/4.\) In Section 1.2 we will discuss methods that permit us to calculate confidently, rather than guess, the values of such limits.

One-Sided Limits

One-sided limits are tools that extend our ability to describe functions. A left-hand limit gives the value to which \(f(x)\) tends as \(x\) is made close to \(a\) from strictly the left side; that is, we consider only \(x \lt a.\) We denote a left-hand limit using the notation \[\lim_{x \to a^-} f(x) = L \cma\] where the negative superscript on \(a^-\) suggests that \(x \lt a.\) (See Figure 3A.) Similarly, a right-hand limit is the value \(f(x)\) approaches as \(x\) is made close to \(a\) from solely the right side, so we consider only \(x \gt a.\) A right-hand limit is written as \[\lim_{x \to a^+} f(x) = L \cma\] for which the positive superscript on \(a^+\) implies that \(x \gt a.\) (See Figure 3B.) Note that we say \(x \lt a\) instead of \(x \leq a,\) and \(x \gt a\) instead of \(x \geq a.\)

Existence of a Limit A limit exists if and only if its one-sided limits exist and equal each other. Mathematically, \(\lim_{x \to a} f(x)\) \(= L\) is true if and only if \[\lim_{x \to a^-} f(x) = L \and \lim_{x \to a^+} f(x) = L \pd\] In other words, we must be able to trace a curve \(f\) near \(a\) and see the values of \(f(x)\) approach \(L\) from both sides of \(x = a.\) If this agreement isn't present, then the limit doesn't exist. For the graphs in Figure 3A and Figure 3B, the one-sided limits of \(f\) at \(a\) are different. Thus, for each graph \(\lim_{x \to a} f(x)\) does not exist.

In Figure 4, determine the values of the following expressions if they exist.

1. \(\ds \lim_{x \to 0} f(x)\)
2. \(\ds \lim_{x \to 2} f(x)\)
3. \(\ds \lim_{x \to 3} f(x)\)

Observe that \(\abs x\) is equivalent to the piecewise function \[\abs x = \bc -x &x \lt 0 \nl x &x \geq 0 \pd \ec \] So \[ \ba \frac{\abs x}{x} &= \bc -x/x &x \lt 0 \nl x/x &x \gt 0 \ec \nl &= \bc -1 &x \lt 0 \nl 1 &x \gt 0 \pd \ec \nl \ea \] Note that \(\abs{x}/x\) is undefined at \(x = 0.\) (See Figure 5.) We find \[\lim_{x \to 0^-} \frac{\abs x}{x} = -1 \and \lim_{x \to 0^+} \frac{\abs x}{x} = 1 \pd\] Since these one-sided limits aren't equal, the limit \(\lim_{x \to 0} \par{\abs{x}/x}\) does not exist.

We construct a table with some values of \(x,\) as follows.

\(x\)	\(\ds \frac{2}{\pi}\)	\(\ds \frac{2}{3\pi}\)	\(\ds \frac{2}{5\pi}\)	\(\ds \frac{2}{7\pi}\)	\(\ds \frac{2}{9\pi}\)	\(\ds \frac{2}{11\pi}\)
\(\ds \sin \frac{1}{x}\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)

Hence, \(\sin(1/x)\) continually oscillates as \(x\) is made closer to \(0.\) Figure 6 shows the graph of \(y = \sin(1/x)\) making endless waves as \(x \to 0.\) The function therefore does not settle to any final value as \(x\) is made close to \(0,\) meaning the limit \(\lim_{x \to 0} \sin(1/x)\) does not exist.

The preceding three examples show instances when a limit \(\lim_{x \to a} f(x)\) may fail to exist. A limit fails to exist if the one-sided limits of \(f\) at \(a\) don't agree with each other. Or a limit can fail to exist if \(f(x)\) doesn't settle to a final value—for example, \(f\) oscillates (such as in Example 5)—as \(x\) is made close to \(a.\)

Limits with Infinity

A function \(f(x)\) may approach some value \(L\) as \(x\) is made closer to some number \(a.\) But what if we let \(x\) increase or decrease without bound? In such cases, we aim to determine whether \(f(x)\) tends to a fixed number as \(x \to \infty\) or as \(x \to -\infty.\) We call these matters limits at infinity.

In general, the notation \[\lim_{x \to \infty} f(x) = L\] means that \(f(x)\) approaches some number \(L\) as we make \(x\) arbitrarily large (as large as we wish)—in other words, as \(x\) increases without bound. Similarly, writing \[\lim_{x \to -\infty} f(x) = L\] indicates that \(f(x)\) tends to \(L\) if we let \(x\) decrease without bound.

We interpret the limit statement as the following question: what happens to \(1/x\) if we let \(x\) grow bigger and bigger—that is, as \(x\) increases without bound? A table of values for \(1/x\) is shown below.

\(x\)	\(\ds \frac{1}{x}\)
\(1\)	\(1\)
\(5\)	\(0.2\)
\(10\)	\(0.1\)
\(100\)	\(0.01\)
\(1000\)	\(0.001\)
\(10000\)	\(0.0001\)
\(100000\)	\(0.00001\)

As we increase \(x,\) the value \(1/x\) shrinks to \(0\) because the number \(1\) is divided by a larger and larger value. If we trace the graph of \(y = 1/x\) (Figure 7) in the direction of increasing \(x,\) then we observe that the curve plunges toward the \(x\)-axis. So \[\lim_{x \to \infty} \frac{1}{x} = \boxed 0\]

Infinite Limits The notation \[\lim_{x \to a} f(x) = \infty\] means that the values of \(f(x)\) become bigger and bigger—increasing without bound—as \(x\) is made closer and closer to \(a.\) We may choose to say a limit equals infinity instead of writing does not exist, which provides less information about a function's behavior. The former statement concedes that a limit doesn't exist while communicating that the function approaches \(\infty.\) Likewise, writing \[\lim_{x \to a} f(x) = -\infty\] asserts that the values of \(f(x)\) decrease without bound as \(x\) approaches \(a.\) This statement suggests that the limit fails to exist because \(f(x)\) approaches negative infinity. In either definition, the value of \(f(a)\) (or whether it is undefined) is irrelevant.

Now let's merge the ideas of limits at infinity with infinite limits: The family of functions \(x^k,\) where \(k\) is a positive constant, satisfies \begin{equation} \lim_{x \to \infty} x^k = \infty \pd \label{eq:lim-x^k} \end{equation} This formula is intuitive because if \(x\) is large, then \(x^k\) is also large. Hence, \(x^k\) grows with \(x.\) Accordingly, the fraction \(1/x^k\) shrinks to \(0\) due to the division of \(1\) by an increasingly large number. Thus, \begin{equation} \lim_{x \to \infty} \frac{1}{x^k} = 0 \pd \label{eq:lim-1/x^k} \end{equation} In Example 6 we used this logic to verify \(\eqref{eq:lim-1/x^k}\) for \(k = 1.\)

CAUTION Understand that \(\infty\) is not a number. Saying a limit equals \(\infty\) or \(-\infty\) simply describes how the limit fails to exist, therefore providing us with more information about the function's behavior. We will elaborate on limits with infinity in Section 1.6.

The following table shows values of \(\sin x\) for selected increasing values of \(x.\)

\(x\)	\(\ds \frac{\pi}{2}\)	\(\ds \frac{3\pi}{2}\)	\(\ds \frac{5\pi}{2}\)	\(\ds \frac{7\pi}{2}\)	\(\ds \frac{9\pi}{2}\)	\(\ds \frac{11\pi}{2}\)
\(\ds \sin x\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)

Observe that as \(x\) is made larger and larger, the sine function continues oscillating between \(-1\) and \(1,\) never settling to any final value (Figure 8). Accordingly, the limit \(\lim_{x \to \infty} \sin x\) does not exist.

More generally, the limit as \(x \to \infty\) or \(x \to -\infty\) of any of the six trigonometric functions \((\sin x,\) \(\cos x,\) \(\tan x,\) \(\sec x,\) \(\csc x,\) and \(\cot x)\) does not exist. These graphs are periodic, meaning they constantly repeat themselves in regular intervals. Thus, none of these functions settles to a final value as \(x\) increases or decreases without bound.

As \(x\) is made closer to \(0,\) the function \(1/x^2\) grows because \(1\) is divided by a progressively smaller number, as shown by the following table.

\(x\)	\(\ds \frac{1}{x^2}\)
\(-1\)	\(1\)
\(-0.1\)	\(100\)
\(-0.01\)	\(10000\)
\(-0.001\)	\(1000000\)
\(-0.0001\)	\(100000000\)

\(x\)	\(\ds \frac{1}{x^2}\)
\(0.0001\)	\(100000000\)
\(0.001\)	\(1000000\)
\(0.01\)	\(10000\)
\(0.1\)	\(100\)
\(1\)	\(1\)

In fact, the values of \(1/x^2\) can be made arbitrarily large by making \(x\) sufficiently small, as shown by the graph in Figure 9. Thus, we say \[\lim_{x \to 0} \frac{1}{x^2} = \boxed{\infty}\] Alternatively, since \(1/x^2\) does not approach a final value as \(x \to 0,\) it is equally correct to say the limit does not exist.

Example 9 shows that \(y = 1/x^2\) grows without bound as \(x\) is made closer to \(0.\) Specifically, \(y = 1/x^2\) has a vertical asymptote at \(x = 0\) because the curve becomes unbounded as \(x\) gets closer to \(0.\) Generally, a function \(f(x)\) has a vertical asymptote at \(x = p\) if any of the following is true: \[ \baat{2} \lim_{x \to p^-} f(x) &= \infty \lspace &&\lim_{x \to p^+} f(x) = \infty \nl \lim_{x \to p^-} f(x) &= -\infty \lspace &&\lim_{x \to p^+} f(x) = -\infty \pd \eaat \] In Figure 10, observe that the graph of \(y = 1/(x - p)\) has a vertical asymptote at \(x = p\) because \(\lim_{x \to p^-} f(x) = -\infty\) and \(\lim_{x \to p^+} f(x) = \infty.\)

The graphs of many functions feature vertical asymptotes. The natural logarithmic function \(y = \ln x,\) for example, has a vertical asymptote at the \(y\)-axis (that is, at \(x = 0\)) because \(\lim_{x \to 0^+} \ln x = -\infty.\) (See Figure 11.) Moreover, the graph of the periodic function \(y = \tan x\) has vertical asymptotes at \(x = \pi/2,\) \(x = 3 \pi/2, \dots\) as well as at \(x = -\pi/2,\) \(x = -3 \pi/2, \dots.\) Generally, all the vertical asymptotes are the lines \(x = \pi/2 + n \pi,\) where \(n\) is an integer. (See Figure 12.) Rational functions have vertical asymptotes at any points where the denominator is \(0\) and the numerator is nonzero.

Let's examine the behavior of \(f(x)\) on both sides of \(x = 4.\) The graph of \(y = f(x)\) has a vertical asymptote at \(x = 4\) because at \(x = 4,\) the denominator is \(0\) and the numerator is nonzero.

Left-Hand Limit If \(x\) is close to \(4\) but smaller than \(4,\) then the denominator \((x - 4)\) is a small negative number while the numerator \(3x\) remains positive. Accordingly, the fraction \(f(x) = 3x/(x - 4)\) is a large negative number. (A positive number is divided by a small negative number.) For example, \(f(3.9) = -117\) and \(f(3.99) = -1197.\) Thus, \(f(x)\) approaches \(-\infty\) as \(x \to 4^-,\) so we write \[\lim_{x \to 4^-} f(x) = \boxed{-\infty}\]

Right-Hand Limit For \(x\) close to \(4\) but bigger than \(4,\) the denominator \((x - 4)\) is a small positive number. At the same time, the numerator \(3x\) is positive. Hence, \(f(x) = 3x/(x - 4)\) is a large positive number. (A positive number is divided by a small positive number.) As an example, \(f(4.1) = 123\) and \(f(4.01) = 1203.\) The pattern is clear: As \(x \to 4^+,\) \(f(x)\) approaches \(\infty.\) So \[\lim_{x \to 4^+} f(x) = \boxed{\infty}\] (See Figure 13.)