APPLICATIONS OF DIFFERENTIATION

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APPLICATIONS OF DIFFERENTIATION
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APPLICATIONS OF DIFFERENTIATION

• In Sections 2.2 and 2.4, we investigated infinite
  limits and vertical asymptotes.
• There, we let x approach a number.
• The result was that the values of y became
  arbitrarily large (positive or negative).

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APPLICATIONS OF DIFFERENTIATION

• In this section, we let become x arbitrarily
  large (positive or negative) and see what happens
  to y.
• We will find it very useful to consider this
  so-called end behavior when sketching graphs.

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APPLICATIONS OF DIFFERENTIATION
4.4Limits at Infinity Horizontal Asymptotes
In this section, we will learn about Various
aspects of horizontal asymptotes.
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HORIZONTAL ASYMPTOTES

• Lets begin by investigating the behavior
• of the function f defined by
• as x becomes large.

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HORIZONTAL ASYMPTOTES

• The table gives values of this
• function correct to six decimal
• places.
• The graph of f has been
• drawn by a computer in the
• figure.

Figure 4.4.1, p. 230
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HORIZONTAL ASYMPTOTES

• As x grows larger and larger,
• you can see that the values of
• f(x) get closer and closer to 1.
• It seems that we can make the values of f(x) as
  close as we like to 1 by taking x sufficiently
  large.

Figure 4.4.1, p. 230
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HORIZONTAL ASYMPTOTES

• This situation is expressed symbolically
• by writing
• In general, we use the notation
• to indicate that the values of f(x) become
• closer and closer to L as x becomes larger
• and larger.

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HORIZONTAL ASYMPTOTES
1. Definition

• Let f be a function defined on some
• interval .
• Then,
• means that the values of f(x) can be
• made arbitrarily close to L by taking x
  
• sufficiently large.

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HORIZONTAL ASYMPTOTES

• Another notation for is
• as
• The symbol does not represent a number.
• Nonetheless, the expression is
  often read asthe limit of f(x), as x
  approaches infinity, is Lor the limit of f(x),
  as x becomes infinite, is Lor the limit of
  f(x), as x increases without bound, is L

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HORIZONTAL ASYMPTOTES

• The meaning of such phrases is given
• by Definition 1.
• A more precise definitionsimilar to
• the definition of Section 2.4is
• given at the end of this section.

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HORIZONTAL ASYMPTOTES

• Geometric illustrations of Definition 1
• are shown in the figures.
• Notice that there are many ways for the graph of
  f to approach the line y L (which is called a
  horizontal asymptote) as we look to the far right
  of each graph.

Figure 4.4.2, p. 231
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HORIZONTAL ASYMPTOTES

• Referring to the earlier figure, we see that,
• for numerically large negative values of x,
• the values of f(x) are close to 1.
• By letting x decrease through negative values
  without bound, we can make f(x) as close as we
  like to 1.

Figure 4.4.1, p. 231
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HORIZONTAL ASYMPTOTES

• This is expressed by writing
• The general definition is as follows.

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HORIZONTAL ASYMPTOTES
2. Definition

• Let f be a function defined on some
• interval .
• Then,
• means that the values of f(x) can be
• made arbitrarily close to L by taking x
• sufficiently large negative.

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HORIZONTAL ASYMPTOTES

• Again, the symbol does not
• represent a number.
• However, the expression
• is often read as
• the limit of f(x), as x approaches
• negative infinity, is L

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HORIZONTAL ASYMPTOTES

• Definition 2
• is illustrated in
• the figure.
• Notice that the graph approaches the line y L
  as we look to the far left of each graph.

Figure 4.4.3, p. 232
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HORIZONTAL ASYMPTOTES
3. Definition

• The line y L is called a horizontal
• asymptote of the curve y f(x) if either

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HORIZONTAL ASYMPTOTES
3. Definition

• For instance, the curve illustrated in
• the earlier figure has the line y 1 as
• a horizontal asymptote because

Figure 4.4.1, p. 230
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HORIZONTAL ASYMPTOTES

• The curve y f(x) sketched here has both y -1
  and y 2 as horizontal asymptotes.
• This is because

Figure 4.4.4, p. 232
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HORIZONTAL ASYMPTOTES
Example 1

• Find the infinite limits, limits at infinity,
• and asymptotes for the function f whose
• graph is shown in the figure.

Figure 4.4.5, p. 232
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HORIZONTAL ASYMPTOTES
Example 1

• We see that the values of f(x) become
• large as from both sides.
• So,

Figure 4.4.5, p. 232
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HORIZONTAL ASYMPTOTES
Example 1

• Notice that f(x) becomes large negative
• as x approaches 2 from the left, but large
• positive as x approaches 2 from the right.
• So,
• Thus, both the lines x -1 and x 2 are
  vertical asymptotes.

Figure 4.4.5, p. 232
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HORIZONTAL ASYMPTOTES
Example 1

• As x becomes large, it appears that f(x)
• approaches 4.
• However, as x decreases through negative
• values, f(x) approaches 2.
• So, and
• This means that both y 4 and y 2 are
  horizontal asymptotes.

Figure 4.4.5, p. 232
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HORIZONTAL ASYMPTOTES
Example 2

• Find and
• Observe that, when x is large, 1/x is small.
• For instance,
• In fact, by taking x large enough, we can make
  1/x as close to 0 as we please.
• Therefore, according to Definition 1, we have

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HORIZONTAL ASYMPTOTES
Example 2

• Similar reasoning shows that, when x
• is large negative, 1/x is small negative.
• So, we also have
• It follows that the line y 0 (the x-axis) is a
  horizontal asymptote of the curve y 1/x.
• This is an equilateral hyperbola.

Figure 4.4.6, p. 233
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HORIZONTAL ASYMPTOTES

• Most of the Limit Laws given
• in Section 2.3 also hold for limits
• at infinity.
• It can be proved that the Limit Laws (with the
  exception of Laws 9 and 10) are also valid if
  is replaced by or .
• In particular, if we combine Laws 6 and 11 with
  the results of Example 2, we obtain the following
  important rule for calculating limits.

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HORIZONTAL ASYMPTOTES
4. Theorem

• If r gt 0 is a rational number, then
• If r gt 0 is a rational number such that xr
• is defined for all x, then

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HORIZONTAL ASYMPTOTES
Example 3

• Evaluate
• and indicate which properties of limits
• are used at each stage.
• As x becomes large, both numerator and
  denominator become large.
• So, it isnt obvious what happens to their ratio.
  
• We need to do some preliminary algebra.

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HORIZONTAL ASYMPTOTES
Example 3

• To evaluate the limit at infinity of any rational
  
• function, we first divide both the numerator
• and denominator by the highest power of x
• that occurs in the denominator.
• We may assume that , since we are
  interested in only large values of x.

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HORIZONTAL ASYMPTOTES
Example 3

• In this case, the highest power of x in the
• denominator is x2. So, we have

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HORIZONTAL ASYMPTOTES
Example 3
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HORIZONTAL ASYMPTOTES
Example 3

• A similar calculation shows that the limit
• as is also
• The figure illustrates the results of these
  calculations by showing how the graph of the
  given rational function approaches the
  horizontal asymptote

Figure 4.4.7, p. 234
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HORIZONTAL ASYMPTOTES
Example 4

• Find the horizontal and vertical
• asymptotes of the graph of the
• function

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HORIZONTAL ASYMPTOTES
Example 4

• Dividing both numerator and denominator
• by x and using the properties of limits,
• we have

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HORIZONTAL ASYMPTOTES
Example 4

• Therefore, the line is
• a horizontal asymptote of the graph of f.

Figure 4.4.8, p. 235
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HORIZONTAL ASYMPTOTES
Example 4

• In computing the limit as ,
• we must remember that, for x lt 0,
• we have
• So, when we divide the numerator by x, for x lt 0,
  we get
• Therefore,

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HORIZONTAL ASYMPTOTES
Example 4

• Thus, the line is also
• a horizontal asymptote.

Figure 4.4.8, p. 235
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HORIZONTAL ASYMPTOTES
Example 4

• A vertical asymptote is likely to occur
• when the denominator, 3x - 5, is 0,
• that is, when
• If x is close to and , then the
  denominator is close to 0 and 3x - 5 is
  positive.
• The numerator is always positive,
  so f(x) is positive.
• Therefore,

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HORIZONTAL ASYMPTOTES
Example 4

• If x is close to but , then 3x 5 lt
  0, so f(x) is large negative.
• Thus,
• The vertical asymptote is

Figure 4.4.8, p. 235
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HORIZONTAL ASYMPTOTES
Example 5

• Compute
• As both and x are large when x is
  large, its difficult to see what happens to
  their difference.
• So, we use algebra to rewrite the function.

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HORIZONTAL ASYMPTOTES
Example 5

• We first multiply the numerator and
• denominator by the conjugate radical
• The Squeeze Theorem could be used to show that
  this limit is 0.

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HORIZONTAL ASYMPTOTES
Example 5

• However, an easier method is to divide
• the numerator and denominator by x.
• Doing this and using the Limit Laws, we obtain

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HORIZONTAL ASYMPTOTES
Example 5

• The figure illustrates this
• result.

Figure 4.4.9, p. 235
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HORIZONTAL ASYMPTOTES
Example 6

• Evaluate
• If we let t 1/x, then as
  .
• Therefore, .

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HORIZONTAL ASYMPTOTES
Example 7

• Evaluate
• As x increases, the values of sin x oscillate
  between 1 and -1 infinitely often.
• So, they dont approach any definite number.
• Thus, does not exist.

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INFINITE LIMITS AT INFINITY

• The notation is used to
• indicate that the values of f(x) become
• large as x becomes large.
• Similar meanings are attached to the following
  symbols

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INFINITE LIMITS AT INFINITY
Example 8

• Find and
• When x becomes large, x3 also becomes large.
• For instance,
• In fact, we can make x3 as big as we like by
  taking x large enough.
• Therefore, we can write

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INFINITE LIMITS AT INFINITY
Example 8

• Similarly, when x is large negative, so is x3.
• Thus,
• These limit statements can also be seen from the
  graph of y x3 in the figure.

Figure 4.4.10, p. 236
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INFINITE LIMITS AT INFINITY
Example 9

• Find
• It would be wrong to write
• The Limit Laws cant be applied to infinite
  limits because is not a number (
  cant be defined).
• However, we can write
• This is because both x and x - 1 become
  arbitrarily large and so their product does too.

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INFINITE LIMITS AT INFINITY
Example 10

• Find
• As in Example 3, we divide the numerator and
  denominator by the highest power of x in the
  denominator, which is just x
• because and
  as

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INFINITE LIMITS AT INFINITY

• The next example shows that, by using
• infinite limits at infinity, together with
• intercepts, we can get a rough idea of the
• graph of a polynomial without computing
• derivatives.

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INFINITE LIMITS AT INFINITY
Example 11

• Sketch the graph of
• by finding its intercepts and its limits
• as and as
• The y-intercept is f(0) (-2)4(1)3(-1) -16
• The x-intercepts are found by setting y 0 x
  2, -1, 1.

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INFINITE LIMITS AT INFINITY
Example 11

• Notice that, since (x - 2)4 is positive,
• the function doesnt change sign at 2.
• Thus, the graph doesnt cross the x-axis
• at 2.
• It crosses the axis at -1 and 1.

Figure 4.4.11, p. 237
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INFINITE LIMITS AT INFINITY
Example 11

• When x is large positive, all three factors
• are large, so
• When x is large negative, the first factor
• is large positive and the second and third
• factors are both large negative, so

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INFINITE LIMITS AT INFINITY
Example 11

• Combining this information,
• we give a rough sketch of the graph
• in the figure.

Figure 4.4.11, p. 237
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PRECISE DEFINITIONS
5. Definition

• Definition 1 can be stated precisely as
• follows.
• Let f be a function defined on some interval
• (a, ).
• Then, means that, for every ,
  
• there is a corresponding number N such that
• if x gt N, then

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PRECISE DEFINITIONS

• In words, this says that the values of f(x) can
• be made arbitrarily close to L (within a
• distance , where is any positive number)
• by taking x sufficiently large (larger than N,
• where N depends on ).

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PRECISE DEFINITIONS

• Graphically, it says that, by choosing x large
• enough (larger than some number N), we can
• make the graph of f lie between the given
• horizontal lines and
• This must be true no matter how small we choose
  .

Figure 4.4.12, p. 238
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PRECISE DEFINITIONS

• This figure shows that, if a smaller value
• of is chosen, then a larger value of N
• may be required.

Figure 4.4.13, p. 238
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PRECISE DEFINITIONS
6. Definition

• Similarly, a precise version of Definition 2
• is given as follows.
• Let f be a function defined on some interval
• ( ,a).
• Then, means that, for every ,
  
• there is a corresponding number N such that,
• if x lt N, then

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PRECISE DEFINITIONS

• This is illustrated in the
• figure.

Figure 4.4.14, p. 238
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PRECISE DEFINITIONS

• In Example 3, we calculated that
• In the next example, we use
• a graphing device to relate this statement
• to Definition 5 with and .

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PRECISE DEFINITIONS
Example 12

• Use a graph to find a number N
• such that, if x gt N, then
• We rewrite the given inequality as

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PRECISE DEFINITIONS
Example 12

• We need to determine the values of x
• for which the given curve lies between
• the horizontal lines y 0.5 and y 0.7
• So, we graph the curve and these lines in the
  figure.

Figure 4.4.15, p. 239
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PRECISE DEFINITIONS
Example 12

• Then, we use the cursor to estimate
• that the curve crosses the line y 0.5
• when
• To the right of this number, the curve stays
  between the lines y 0.5 and y 0.7

Figure 4.4.15, p. 239
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PRECISE DEFINITIONS
Example 12

• Rounding to be safe, we can say that,
• if x gt 7, then
• In other words, for , we can choose N
  7 (or any larger number) in Definition 5.

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PRECISE DEFINITIONS
Example 13

• Use Definition 5 to prove
• Given , we want to find N such that, if
  x gt N, then
• In computing the limit, we may assume that x gt 0
• Then,

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PRECISE DEFINITIONS
Example 13

• Lets choose
• So, if , then
• Therefore, by Definition 5,

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PRECISE DEFINITIONS
Example 13

• The figure illustrates the proof by
• showing some values of and the
• corresponding values of N.

Figure 4.4.16, p. 239
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PRECISE DEFINITIONS
7. Definition

• Finally, we note that an infinite limit at
  infinity
• can be defined as follows.
• Let f be a function defined on some interval
• (a, ).
• Then, means that, for every
• positive number M, there is a corresponding
• positive number N such that,
• if x gt N, then f(x) gt M

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PRECISE DEFINITIONS

• The geometric illustration is
• given in the figure.
• Similar definitions apply when the symbol is
  replaced by

Figure 4.4.17, p. 240