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Functions and single-variable calculus A course resource for Math 218

Ellen Goldstein, Aaron Greicius

Section 1.11 Introduction to limits

Subsection Informal definition of the limit

Many of the questions we will study in regard to functions revolve around how the output of a function \(f\) changes with respect to its input. As such we want to develop useful and precise language and tools for describing this relation. The limit of a function, defined informally here and rigorously later, is one such tool. Before getting to its informal definition, we illustrate with a graphical example the type of statements we wish to formalize with limit language.
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Example 1.11.1. Function behavior.

Suppose \(f\) is the function whose graph is given below.
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Graph of complicated function
Figure 1.11.2. Graph of function \(f\)
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Describe how the output \(f(x)\) behaves as the input \(x\) approaches each of the values \(a,b,c,d,e\text{.}\) For some of these values, you might consider how \(x\) approaches from the left and right separately.
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Solution.
As \(x\) approaches \(a\text{,}\) either from the left or right, the values \(f(x)\) of the function get arbitrarily large and negative.
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As \(x\) approaches \(b\) from the left, the values \(f(x)\) of \(f\) get closer and closer to \(0\text{.}\) However, as \(x\) approaches \(b\) from the right, the values \(f(x)\) get closer and closer to \(A\text{.}\)
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As \(x\) approaches \(c\) from the left or right, the values \(f(x)\) of the function approach the real number \(M\text{.}\) We also happen to have \(f(c)=B\text{:}\) i.e., the value of \(f\) at \(x=c\) is equal to \(B\text{.}\)
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As \(x\) approaches \(d\) from the left, the values \(f(x)\) of the function approach \(C\text{.}\) However, as \(x\) approaches \(d\) from the right, the values \(f(x)\) oscillate wildly above and below \(C\) and do not seem to approach any single value.
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As \(x\) approaches \(e\) from the left or right, the values \(f(x)\) of the function approach \(D\text{.}\) Interestingly, however, the actual value of \(f\) at the input \(x=e\) is not equal to \(D\text{.}\)
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As Example 1.11.1 illustrates, an English description of how a function behaves for inputs near a given value can be quite a mouthful. Consider the limit notation introduced below as a means of making these descriptions much more concise. Roughly speaking, we will write
\begin{equation*} \lim\limits_{x\to a}f(x)=L \end{equation*}
to say that the values of \(f\) approach \(L\) as the input \(x\) appraoches \(a\text{.}\) This is the basic gist of Definition 1.11.3, though it takes a little more care in the setup.
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Definition 1.11.3. Limit (informal).

Suppose \(f\) is a function defined everywhere on an open interval containing the point \(a\in \R\text{,}\) except possibly at \(a\) itself. We say that the limit of \(f\) as \(x\) approaches \(a\) exists if there is a value \(L\) such that the function value \(f(x)\) can be made arbitrarily close to \(L\) provided \(x\) is sufficiently close (but not equal) to \(a\text{.}\)
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When this is the case, we call \(L\) the limit of \(f\) as \(x\) approaches \(a\) and write
\begin{equation} \lim_{x\to a}f(x)=L\text{.}\tag{1.68} \end{equation}
When the limit does not exist we will say that \(\lim\limits_{x\to a}f(x)\) does not exist.
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Remark 1.11.4. Limit definition (informal).

You might be thinking that our so-called informal limit definition has quite a formal ring to it. What prevents it from being a rigorous, formal definition? The issue lies precisely in making mathematical sense of the phrases “arbitrarily close” and “sufficiently close”. As intuitive as those notions may seem, we need a precise mathematical definition of them if we have any hope of building a theory on solid foundations. That is the motivation behind our formal definition Definition 1.14.1, which we will meet soon.
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Example 1.11.5. Limit description of graph.

Let \(f\) be the function with graph given in Figure 1.11.2. Use the language and notation of limits to describe the behavior of \(f\) for inputs \(x\) near the values \(a,b,c,d,e\text{.}\)
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Solution.
The limit \(\lim\limits_{x\to a}f(x)\) does not exist. There is no value \(L\) that \(f(x)\) approaches for inputs \(x\) sufficiently close to \(a\text{.}\) Indeed, as \(x\) gets arbitrarily close to \(a\text{,}\) \(f(x)\) becomes arbitrarily large and negative.
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The limit \(\lim\limits_{x\to b}f(x)\) does not exist. If \(x\) is arbitrarily close to \(b\text{,}\) and less than \(b\) (to the left), then the values \(f(x)\) gets arbitrarily close to \(0\text{;}\) thus if the limit existed it would have to be equal to \(0\text{.}\) And yet for inputs \(x\) arbitrarily close to \(b\) but greater than \(b\) (to the right), the values \(f(x)\) are greater than \(1\text{:}\) i.e., they get no closer than a distance of \(1\) from the value \(0\text{.}\) This implies \(0\) is not the limit, and hence that no limit exists.
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We have \(\lim\limits_{x\to c}f(x)=B\text{.}\) Since \(B=f(c)\text{,}\) we have \(\lim\limits_{x\to c}f(x)=f(c)\) in this case.
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The limit \(\lim\limits_{x\to d}f(x)\) does not exist. See the explanation in Example 1.11.1.
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We have \(\lim\limits_{x\to e}f(x)=D\text{.}\) Note that in this case \(e\) is an element of the domain and \(\lim\limits_{x\to e}f(x)\ne f(e)\text{.}\)
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Example 1.11.6. Limit as compared to value.

Provide graphs of the following functions on their entire implied domain:
\begin{align*} f(x) \amp =\frac{x^2-1}{x-1} \amp g(x)\amp=\begin{cases} x+1 \amp \text{if } x\ne 1\\ 1 \amp \text{if } x=1 \end{cases} \amp h(x)\amp =x+1\text{.} \end{align*}
For each function discuss the limit behavior at \(x=1\) as compared with the value of the function at \(x=1\text{.}\)
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Solution.
The domain of \(f\) is \(\{x\in \R\mid x\ne 0\}\text{.}\) Since \(f(x)=(x-1)(x+1)/(x-1)=x+1\) for all \(x\ne 1\text{,}\) the graph of \(f\) is the line \(y=x+1\) for all \(x\ne 1\text{.}\) Similar reasoning for \(g\) and \(h\) yields the graphs below.
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Graph of f
Figure 1.11.7. Graph of \(f\)
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Graph of g
Figure 1.11.8. Graph of \(g\)
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Graph of h
Figure 1.11.9. Graph of \(h\)
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The basic idea behind the limit is to observe how the outputs \(f(x)\) change as the input \(x\) approaches some value \(a\text{,}\) written \(x\to a\) for short. As we saw above, a detailed graph can give a strong indication of whether the values \(f(x)\) approach some fixed value \(L\text{.}\) This type of behavior can also be made evident by a sufficiently detailed table of values.
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Example 1.11.10. Limits: using table of values.

Consider the function
\begin{equation*} h(x) = \frac{x-3}{x^3 - 3 x^2 - 2 x + 6}\text{.} \end{equation*}
You can verify for your self that \(3\) is a root of the denominator of \(h\text{,}\) and hence that \(h\) is not defined at \(x=3\text{.}\)
  1. What does the table of values below suggest about \(\lim_{x\to 3}h(x)\text{?}\) Do you think the limit exists? If so, estimate its value.
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    Table 1.11.11. Table of values of \(h\)
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    \(x\) \(2.9\) \(2.99\) \(2.999\) \(3\) \(3.001\) \(3.01\) \(3.1\)
    \(h(x)\) \(0.156\) \(0.144\) \(0.14298\) \(*\) \(0.14273\) \(0.142\) \(0.131\)
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  2. Perform some algebra on \(h\) to get a better idea of what \(h(x)\) is for \(x\ne 3\text{.}\) Does this algebra suggest a potential exact value for \(\lim\limits_{x\to 3}h(x)\text{?}\)
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Solution.
  1. First note that the table provides values \(h(x)\) for inputs \(x\) that get reasonably close to \(3\text{:}\) from the left, we have the sequence of inputs \(2.9, 2.99, 2.999\text{,}\) and from the right we have the sequence \(3.1, 3.01, 3.001\text{.}\) The closest inputs included in the table are \(2.999\) and \(3.001\text{,}\) which are a distance of
    \begin{equation*} .001=\abs{3.0001-3}=\abs{2.999-3} \end{equation*}
    away from \(3\text{.}\) Moreover, we notice that as these inputs get closer to \(3\text{,}\) the outputs \(h(x)\) seem to approach a value between \(0.142\) and \(0.143\text{.}\) Thus, we are inclined to believe that the limit \(\lim\limits_{x\to 3}h(x)\) does indeed exist, and that
    \begin{equation*} 0.142 \leq \lim\limits_{x\to 3}h(x) \leq 0.143 \text{.} \end{equation*}
    In fact, since the values \(h(x)\) decrease as \(x\) approaches from the left, and increase as \(x\) approaches from the right, we might further guess that
    \begin{equation*} 0.14273 \leq \lim\limits_{x\to 3}h(x) \leq 0.14298\text{.} \end{equation*}
    Notice, however, that that guess would be conditioned on the assumption that \(h\) is decreasing for all \(x\) sufficiently close to \(3\) on the left, and all \(x\) sufficiently close to \(x\) on the right; and this is an assumption we are not in a position to verify at the moment.
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  2. We can factor the denominator of \(h\) as
    \begin{equation*} x^3 - 3 x^2 - 2 x + 6 = (x-3)(x^2-2)\text{,} \end{equation*}
    which means that we have
    \begin{equation*} h(x)=\frac{1}{x^2-2} \end{equation*}
    for all \(x\ne 3\) in the domain of \(h\text{.}\) Let \(g(x)=\frac{1}{x^2-2}\text{.}\) Since \(h(x)=g(x)\) for all \(x\ne 3\text{,}\) the limit behaviors of the two functions are the same at \(x=3\text{.}\) Note that \(g\) is defined at \(x=3\) and satisfies
    \begin{equation*} g(3)=\frac{1}{9-2}=\frac{1}{7}\approx 0.142857\text{.} \end{equation*}
    Wait a minute, that looks a whole lot like our estimate of \(\lim\limits_{x\to a}h(x)\) in part (a)! Are we in a position to say with assurance that
    \begin{equation*} \lim\limits_{x\to 3}h(x)=\frac{1}{7}? \end{equation*}
    Not quite yet, but the evidence is stacking up. We could conclude this if we knew that \(\lim\limits{x\to 3}g(x)=1/7\text{.}\) This seems like it should be true since \(g\) is actually defined at \(x=3\) and we have seen that rational functions behave “nicely” around points in their domain. Soon we will have a more precise way of saying this that will give us the answer we desire.
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Subsection One-sided limits

Let’s look at the limit behavior of the sign function
\begin{equation*} \sgn(x)=\begin{cases} 1\amp \text{if } x> 0 \\ -1\amp \text{if } x< 0\\ 0\amp \text{if } x=0. \end{cases} \end{equation*}
near \(x=0\text{.}\) (See Figure 1.10.2.) Since the function is defined piecwise, it is natural to look at the behavior for \(x> 0\) and \(x< 0\) separately. Since \(f(x)=1\) for all \(x> 0\text{,}\) we conclude that \(f(x)\) approaches \(1\) for all \(x\) sufficiently close to \(0\) and lying to its right. Similarly, we see that \(f(x)\) approaches \(-1\) for all \(x\) sufficiently close to \(0\) and lying to its left. Wouldn’t it be nice if we had some notation to capture this “from the left” and “from the right” behavior? The notion of one-sided limits, defined below, gives us just what we need. In particular, using its notation, we can say more succinctly that
\begin{align*} \lim\limits_{x\to 0^+}\sgn(x) \amp =1\\ \lim\limits_{x\to 0^-}\sgn(x) \amp =-1\text{.} \end{align*}
Before getting to that definition, let’s finish our analysis of the behavior of \(\sgn\) near \(x=0\text{.}\) Since the values of \(\sgn\) approach two distinct values (\(1\) and \(-1\)), depending on whether \(x\) approaches \(0\) from the right or left, we conclude that there is no single value that values \(\sgn(x)\) approach as \(x\) gets sufficiently close to \(0\text{,}\) and hence that \(\lim\limits_{x\to 0}\sgn(x)\) does not exist. (We make this reasoning more official in Theorem 1.11.15.)
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Definition 1.11.12. One-sided limits: informal.

Let \(f\) be a function, and let \(a\in \R\text{.}\)
  1. Left-hand limit.
    Assume \(f\) is defined on an open interval of the form \((c,a)\text{.}\) We say the left-hand limit of \(f\) as \(x\) approaches \(a\) from the left exists if there is a value \(L\) such that \(f(x)\) can be made arbitrarily close to \(L\) provided \(x\) is sufficiently close to \(a\) and lying to its left.
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    When this is the case we call \(L\) the left-hand limit of \(f\) as \(x\) approaches \(a\) from the left, and write
    \begin{equation*} \lim_{x\to a^-}f(x)=L\text{.} \end{equation*}
    When no such \(L\) exists, we say that the left-hand limit \(\lim\limits_{x\to a^-}f(x)\) does not exist.
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  2. Right-hand limit.
    Assume \(f\) is defined on an open interval of the form \((a,d)\text{.}\) We say the right-hand limit of \(f\) as \(x\) approaches \(a\) from the right exists if there is a value \(L\) such that \(f(x)\) can be made arbitrarily close to \(L\) provided \(x\) is sufficiently close to \(a\) and lying to its right.
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    When this is the case we call \(L\) the right-hand limit of \(f\) as \(x\) approaches \(a\) from the right, and write
    \begin{equation*} \lim_{x\to a^+}f(x)=L\text{.} \end{equation*}
    When no such \(L\) exists, we say that the right-hand limit \(\lim\limits_{x\to a^+}f(x)\) does not exist.
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Example 1.11.13. Visualizing one-sided limits.

Suppose \(f\) is the function whose graph is given below.
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Graph of complicated function
Figure 1.11.14. Graph of function \(f\)
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Describe the behavior of \(f\) near the inputs \(a\text{,}\) \(b\text{,}\) \(d\text{,}\) and \(e\) using one-sided limit notation.
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Solution.
  • Neither \(\lim\limits_{x\to a^-}f(x)\) nor \(\lim\limits_{x\to a^+}f(x)\) exists, since the function values get arbitrarily large (and negative) as \(x\) approaches \(a\) from either side.
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  • We have
    \begin{align*} \lim_{x\to b^-}f(x) \amp = 0 \amp \lim_{x\to b^+}f(x)\amp =A\text{.} \end{align*}
    Thus in this case the left- and right-handed limits of \(f\) both exist, but do not agree.
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  • We have \(\lim\limits_{x\to d^-}f(x)=C\text{.}\) By contrast, \(\lim\limits_{x\to d^+}f(x)\) does not exist as the values of \(f\) oscillate wildly as \(x\) approaches \(d\) from the right.
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  • We have
    \begin{equation*} \lim\limits_{x\to e^-}f(x)=\lim\limits_{x\to e^+}f(x)=D\text{.} \end{equation*}
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As was suggested by Example 1.11.13 and made official by the next theorem, \(\lim_{x\to a}f(x)\) exists if and only if both one-sided limits exist and are equal.
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Theorem 1.11.15 is particularly useful in that it gives us a convenient means of showing that a limit does not exist. Let’s apply this theorem to some of our previous examples and see how it simplifies things.
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Looking again at the sign function, we see that the left and right limits do indeed exist, but are not equal: we have
\begin{align*} \lim\limits_{x\to 0^-}\sgn(x)\amp =-1 \\ \lim\limits_{x\to 0^+}\sgn(x) \amp =1 \text{.} \end{align*}
Since \(\lim\limits_{x\to 0^-}\sgn(x)\ne \lim\limits_{x\to 0^+}\sgn(x)\text{,}\) we conclude that \(\lim\limits_{x\to 0}\sgn(x)\) does not exist.
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Now return to the function \(f\) whose graph is given in Example 1.11.13, and let’s see how we can apply Theorem 1.11.15 at some of the limit points. Since
\begin{align*} \lim\limits_{x\to b^-}f(x) \amp = 0 \\ \lim\limits_{x\to b^+}f(x) \amp = A\ne 0 \text{,} \end{align*}
we conclude that \(\lim\limits_{x\to b}f(x)\) does not exist. By contrast since
\begin{align*} \lim\limits_{x\to e^-}f(x) \amp = \lim\limits_{x\to e^+}f(x)=D \text{,} \end{align*}
we conclude that \(\lim\limits_{x\to e}f(x)\) exists and that
\begin{equation*} \lim\limits_{x\to e}f(x)=\lim\limits_{x\to e^-}f(x)=\lim\limits_{x\to e^+}f(x)=D\text{.} \end{equation*}
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We end our section by extending somewhat our definition of the limit. Previously our definition of limit only applied to points \(a\in \R\) for which the function was defined for all \(x\ne a\) on an open interval \((c,d)\) containing \(a\text{.}\) The language of one-sided limits allows us to extend this definition to “endpoints” of the domain, as the next definition makes clear. Theorem 1.11.15 guarantees that this definition is a consistent extension of Definition 1.11.3.
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Definition 1.11.16. Limit at endpoints of domain.

Let \(f\) be a function, and let \(a\in \R\text{.}\)
  1. Right endpoint.
    If \(f\) is defined for all \(x\) on an open interval \((c,a)\text{,}\) but not for all \(x\ne a\) in a full open interval containing \(a\text{,}\) then we say the limit of \(f\) as \(x\) approaches \(a\) exists if \(\lim\limits_{x\to a^-}f(x)\) exists. When this is the case, we define
    \begin{equation*} \lim_{x\to a}f(x)=\lim_{x\to a^-}f(x)\text{.} \end{equation*}
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  2. Left endpoint.
    If \(f\) is defined for all \(x\) on an open interval \((a,d)\text{,}\) but not for all \(x\ne a\) in a full open interval containing \(a\text{,}\) then we say the limit of \(f\) as \(x\) approaches \(a\) exists if \(\lim\limits_{x\to a^+}f(x)\) exists. When this is the case, we define
    \begin{equation*} \lim_{x\to a}f(x)=\lim_{x\to a^+}f(x)\text{.} \end{equation*}
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