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Chapter 1: Limits
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Section 1.1: Naive Limits
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Essentials



A Touch of History


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The need for a clear notion of the mathematical concept of a limit arises when defining the derivative, the essential subject in differential calculus. Roughly speaking, Isaac Newton defined "instantaneous velocity" in physics as the limit of "average velocities". A translation of this into more modern mathematical terms would say that the derivative is the limit of slopes of secant lines. Either way, a ratio is examined at a point where both numerator and denominator become zero, so that the meaningless symbol $0\/0$ results. It was nearly 200 years before the mathematical community finally became comfortable with the idea of a limit.

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The following illustrates one way to begin thinking about the idea of the mathematical limit.

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•

Figure 1.1.1 shows a graph of $f\left(x\right)\=\frac{{x}^{2}1}{x1}$, where the red circle indicates that the point $\left(1\,2\right)$ is missing from the graph of the function. Moreover, except for the red circle, the graph of $f$ seems to be that of the straight line $y\=x\+1$. Why the circle, and why the line?

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The point $\left(1\,2\right)$ is not on the graph of the line because $f\left(1\right)$ is not a number. The function has no value at this point. At $x\=1$, the expression $\frac{{x}^{2}1}{x1}$becomes $\frac{0}{0}$, which is a meaningless collection of symbols. Any real number divided by zero is meaningless, because it is never itself a real number; hence, the gap in the graph where the point $\left(1\,2\right)$ should be, and this is indicated by the red circle.


>

module()
local p;
p[1]:=plot((x^2  1)/(x  1), x = 2 .. 3, discont = [showremovable = [symbol = solidcircle, color = white]], color = black):
p[2] := plot([[1, 2]], style = point, symbol = circle, symbolsize = 20, color = red):
p[3]:=plots:display(p[1],p[2]):
print(p[3]);
end module:


Figure 1.1.1 Graph of $f\left(x\right)\=\frac{{x}^{2}1}{x1}$ showing a small red circle centered at the point $\left(1\,2\right)$






•

But why is the graph of $\frac{{x}^{2}1}{x1}$ nearly the graph of $y\=x\+1$? Well, the numerator factors to $\left(x\+1\right)\left(x1\right)$, so the fraction itself reduces to $x\+1$ except at $x\=1$ where the fraction has no meaning and is not defined. The function $f$ could therefore be represented as the piecewise defined object

$f\left(x\right)\=\left\{\begin{array}{cc}\frac{{x}^{2}1}{x1}& x\ne 1\\ \mathrm{undefined}& x\=1\end{array}\right.\=\left\{\begin{array}{cc}x\+1& x\ne 1\\ \mathrm{undefined}& x\=1\end{array}\right.$
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When $f\left(x\right)$ is simplified to $x\+1$, an expression that has the value 2 when $x\=1$, the modified expression can be used to estimate the value $f$ would have if $f$ were evaluated near $x\=1$. The value 2 is the limiting value for $f$ as $x$ approaches 1.

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In other words, since $f\left(x\right)$ is undefined at $x\=1$, extend the domain of $f$ to include $x\=1$. This results in a new function $g\left(x\right)\=x\+1$ that agrees with $f$ at all points where $f$ is defined, but has a value of $g\left(1\right)\=2$. This value is taken as the limit of $f$ as $x\to 1$.

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The Language and Notation for the Limit Concept


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In everyday speech, the word "limit" usually denotes some sort of boundary. In the calculus, the term is applied to functions, and in a much more precise, but related, sense. It is a "bounding" number to which a function's values get arbitrarily close. In particular, $L$ is the limit of the function $f\left(x\right)$ as $x$ approaches $a$, if the numbers $f\left(x\right)$ become increasingly close to $L$ when $x$ becomes increasingly close to $a$.
The formal way of writing this relationship between, $f\left(x\right)$, $a$, and $L$ is
$\underset{x\to a}{lim}f\left(x\right)\=L$
The word "limit" is captured as an operator "lim" under which "$x\to a$" denotes "$x$ approaches $a$". This operator is applied to the function $f$, resulting in the number $L$. The notation means that the nearer the values of $x$ are to the number $a$, the closer the values of $f\left(x\right)$ are to the number $L$. If $L$ is the limit of $f\left(x\right)$ as $x$ approaches $a$, then the values of the function are very nearly the number $L$ for values of $x$ very nearly the number $a$, but the function itself need not even be defined at $x\=a$.
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If the function is actually defined at $x\=a$ so that $f\left(a\right)$ is a valid number, that number may or may not be the value $L$! The limit at $x\=a$ merely expresses something about the value of the function near $x\=a$, not at $x\=a$.
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This section views the limit naively, the way it was understood prior to the formal definition that evolved more than a century after it was first used in the calculus by Newton and Leibniz. The investigations and algebraic manipulations found in this section are based on the idea that "limit" means "$x$ near $a$ guarantees $f\left(x\right)$ near $L$."
This section uses graphs to estimate a limit, and develops some basic manipulative techniques for finding the actual limiting value $L$. These techniques will be combined to develop the concepts of onesided limits and to support an initial look at two special trigonometric limits. The section concludes with a short reminder that graphs and tablesofvalue can be misleading.
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Estimating Limits from Graphs and Tables


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Many limit calculations are illuminated by graphs and/or a table of values. However, it is easy to devise pathological functions for which such devices fail to provide a correct evaluation of a limit. Insofar as these devices are useful, their application is certainly encouraged. But the ultimate caution remains: only analytic methods are completely trustworthy. See Example 1.1.1 and Example 1.1.2.


Basic Analytic Methods


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The following examples will illustrate two basic techniques for evaluating the limit of an expression for which direct evaluation would generate a divisionbyzero error. Hence, each method applies to fractions, the one to a rational function (the ratio of two polynomials) and the other to fractions that contain the squareroot function. The first technique will be called factor and cancel (see Example 1.1.3); and the other, rationalize the numerator (see Example 1.1.4).
Later, Example 1.1.8 will illustrate Principle 1.1.1, another analytic principle for determining limits of certain functions that need not be fractions.
1. $f\left(x\right)\=g\left(x\right)h\left(x\right)$
2. $\underset{x\to a}{lim}g\left(x\right)\=0$
3. $h\left(x\right)$ is bounded by a positive constant $M$ in a neighborhood of $x\=a$
⇒
$\underset{x\to a}{lim}f\left(x\right)\=0$

Principle 1.1.1 Limit of a product where one factor is bounded



So, if $f$ is a product in which one factor goes to zero, but the other factor is merely bounded, then the limit of the product will be zero.
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PiecewiseDefined Functions and OneSided Limits


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A piecewisedefined function is one whose domain is subdivided, and a different rule is used on each of these different subdivisions. For such a function, obtaining a limit at a point where the subdivisions meet requires obtaining the limits of the separate rules. Such limits are called onesided limits. The function itself has a limit (the twosided limit) at that point only if the two onesided limits exist and have the same value.
For example, if $f\left(x\right)\=\{\begin{array}{cc}g\left(x\right)& x<a\\ h\left(x\right)& x\ge a\end{array}$, then determining $\underset{x\to a}{lim}f\left(x\right)$ requires determining the two onesided limits in Table 1.1.1.
Limit from the Left

Limit from the Right

$\underset{x\to a}{lim}f\left(x\right)\=\underset{x\to a}{lim}g\left(x\right)$

$\underset{x\to a\+}{lim}f\left(x\right)\=\underset{x\to a\+}{lim}h\left(x\right)$

Table 1.1.1 Onesided limits



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The limit from the left (lefthand limit) of $x\=a$ is taken through values of $x$ that are to the left of $a$, and this approach to $a$ is designated by the minus sign appended to the $a$. Of course, this limit is taken of the "lefthand" rule, that is, of the function $g\left(x\right)$.
Similarly, the limit from the right (righthand limit) of $x\=a$ is taken through values of $x$ that are to the right of $a$, and this approach to $a$ is designated by the plus sign appended to the $a$. Clearly, this limit is taken of the "righthand" rule, that is, of the function $h\left(x\right)$. See Example 1.1.5 and Example 1.1.6.
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Limits of Oscillating Functions


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Periodic functions such as the sine and cosine functions oscillate, and have especially interesting properties when composed with other functions. In Example 1.1.7 and Example 1.1.8, limits for such oscillatory functions are examined.


The Transcendental Number e


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The number $\mathrm{\π}$ is irrational because it cannot be represented as the ratio of two integers. Moreover, it is transcendental, that is, it is not the zero of a nonconstant polynomial equation with rational coefficients.
The number
$e\=\underset{h\to 0}{lim}{\left(1\+h\right)}^{1\/h}\doteq 2.7182818284590452354\dots$
is likewise both irrational and transcendental. This limit can be shown to exist, and to lie between 2 and 3, but it can only be approximated numerically. Figure 1.1.4 captures the essence of such numeric calculations in its graph of the function $f\left(h\right)\={\left(1\+h\right)}^{1\/h}$.
The black dot in Figure 1.1.12 is a reminder that $f\left(0\right)$ is the undefined form ${1}^{1\/0}$, but that the limit of $f$ as $h\to 0$, does exist.


Figure 1.1.4 Graph of $f\left(h\right)\={\left(1\+h\right)}^{1\/h}$




Some opine that this number gets its name from the word exponential; but others, that it honors Leonhard Euler who coined the name in 1727.



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The related limit ${e}^{x}\=\underset{h\to 0}{lim}{\left(1\+hx\right)}^{1sol;h}$ defines the exponential function whose base is the "exponential e." To see this, write
$\underset{h\to 0}{lim}{\left(1\+hx\right)}^{1sol;h}$

$\=\underset{H\to 0}{lim}{\left(1\+H\right)}^{x\/H}$

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$\={\left(\underset{H\to 0}{lim}{\left(1\+H\right)}^{1\/H}\right)}^{x}$

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$\={e}^{x}$



where $hxequals;H$, and $H\to 0$ as $h\to 0$.
Implement this number in Maple by selecting the symbol $\ⅇ$ from the Common Symbols palette or from the Constants and Symbols palette. It can also be obtained by typing an ordinary e in Math mode, and invoking Command Completion either from the keyboard (Escape key, or Control + SpaceBar) or from the Tools menu. In the popup generated by Command Completion, select Exponential 'e'. Typing as text an ordinary "e" that simply looks like the exponential e is not sufficient. The exponential e, which in Maple is actually the number $\mathrm{exp}\left(1\right)$, must be inserted by one of the devices delineated above.
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Précis


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Table 1.1.3 summarizes some of the main points of this section.
•

To evaluate $\underset{x\to a}{lim}f\left(x\right)$, see if $f\left(a\right)$ is defined. If it is, that number is the limit.

•

If $f\left(a\right)$ is not defined because of a division by zero, try "factor and cancel" or "rationalize the numerator."

•

Graph $f\left(x\right)$ and/or obtain a table of values on either side of $x\=a$. Use these two tools carefully; they can be misleading.

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If $f\left(x\right)$ is piecewise defined and $a$ is a division point in its domain, take onesided limits.
If both onesided limits exist and are equal, the limit itself is this common value.
If the onesided limits exist but are not equal, the function has a jump at $x\=a$.
If either of the onesided limits does not exist, the limit itself does not exist.

•

Use the two trig limits in Table 1.1.2 as "math facts" for evaluating limits of trig functions.

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The behavior of a highly oscillatory function might be determined from its graph, whether the oscillations remained bounded, grow without bounded, or shrink to zero.


Table 1.1.3 Summary of highlights of Section 1.1



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Examples


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Example 1.1.1

Graph $f\left(x\right)\=\frac{{x}^{4}13{x}^{3}\+62{x}^{2}120x\+64}{x4}$ and thereby estimate $\underset{x\to 1}{lim}f\left(x\right)$ and $\underset{x\to 4}{lim}f\left(x\right)$.

Example 1.1.2

Using a graph and a table of values, estimate $\underset{x\to 0}{lim}g\left(x\right)$, where $g\left(x\right)\=\frac{{\left(100x\mathrm{cos}\left(x\right)\right)}^{2}}{100000}$.

Example 1.1.3

Apply the factor and cancel technique to obtain $\underset{x\to 4}{lim}f\left(x\right)\=8$, where $f\left(x\right)$ is the function in Example 1.1.1.

Example 1.1.4

Apply the technique of rationalizing the numerator to obtain $\underset{x\to 0}{lim}h\left(x\right)$, where $h\left(x\right)\=\frac{\sqrt{1\+x}1}{x}$.

Example 1.1.5

Obtain $\underset{x\to 1}{lim}f\left(x\right)$, if $f$ is the piecewise function given by $f\left(x\right)\=\left\{\begin{array}{cc}2{x}^{2}& x<1\\ \sqrt{x1}\frac{2x}{3}\+2& x\ge 1\end{array}\right.$.

Example 1.1.6

Obtain $\underset{x\to 1}{lim}g\left(x\right)$, if $g$ is the piecewise function given by $g\left(x\right)\=\left\{\begin{array}{cc}2{x}^{2}& x<1\\ \sqrt{x1}xplus;2& x\ge 1\end{array}\right.$.

Example 1.1.7

Evaluate $\underset{x\to 0}{lim}\mathrm{sin}\left(1\/x\right)$.

Example 1.1.8

Evaluate $\underset{x\to 0}{lim}x\mathrm{sin}\left(1sol;x\right)$.



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