The limit

Document Sample
The limit Powered By Docstoc
					                                 Summary of limits, part 11                                         the thing inside the limit [f (x + h) − f (x)]/h is not defined at h = 0 but that doesn’t mean
                                                                                                    that it is usually hard to work out what the limit is; for example, when calculating the
                                                                                                    derivative of f (x) = x2 we have
                                          6 November 2008                                                                  df        x2 + 2xh + h2 − x2
                                                                                                                              = lim                       = lim (2x + h) = 2x                  (7)
   So the purpose of these notes is to bring all the different limits together into one                                    dx2 h→0             h              h→0

summary; limits have a single idea behind them, but there is a set of different definitions           we have divided above and below by h, something that is possible provided h = 0; when,
designed to deal with different circumstances and this can get confusing.                            in fact, h = 0 is the point we are interested in; the fact we are working inside a limit means
                                                                                                    that this isn’t a problem.
                                                                                                        You can think of the limit as a piece of mathematical technology designed to get around
The limit                                                                                           the fact that 0/0 isn’t defined: it finds where the function was headed as it moves towards
                                                                                                    the place where it gives 0/0. Although it might be obvious what the value should be, that
The idea is that if a function, f (x) gets closer and closer to some value L the closer x gets      doesn’t mean we can ignore the need to work inside a limit, without an exact mathematical
to some a then we say the limit of f (x) as x goes to a is L, or,                                   language defining what we mean by ‘going towards’, it would not be possible to prove
Limit: (Informal definition). If the value of f (x) can be made as close as we like to L by          theorems about differenciation. The precise mathematical language is
taking values of x sufficiently close to a but not equal to a then we write                           Limit: (Formal definition)
                                                                                                                                             lim f (x) = L                                     (8)
                                               lim f (x) = L                                  (1)                                                 x→a
                                               x→a
                                                                                                    if and only if for all ǫ > 0 there exists a δ > 0 such that for all |x − a| < δ |f (x) − L| < ǫ.
   The difficulty with the limit is that the examples sometimes seem a little contrived,                  So, in the formal definition we refered to ‘f (x) as close as we like to L’, this is made
but, as we see, having the limit available as a piece of mathematical technology makes              exact by saying that gor any ǫ, that is ∀ǫ > 0, we can make |f (x) − L| < ǫ. Before we said
defining differenciation much easier. The typical example for the limit is a function like            that we can make f (x) as close as we like to L by taking ‘x sufficiently close to a’, here
                                                                                                    we have made this exact, by saying we can find a δ, that is ∃δ > 0 such that if |x − a| < δ
                                                             x2 − x                                 then |f (x) − L| < ǫ. The role of the formal definition is to make the ideas ‘as close as we
                                               f (x) =                                        (2)
                                                             x−1                                    like’ and ‘sufficiently close’ into mathematics that can be used for proving theorems, such
since                                                                                               as the the nice theorem. In this course however, we don’t actually prove this theorem.
                                   x2 − x    x(x − 1)                                                   The niceness theorem is the theorem that says the limit behaves nicely, it says that if
                                          =                                          (3)            limx→a f (x) = L1 and limx→a g(x) = L2 then
                                    x−1       x−1
the x − 1 above and below cancel provided x = 1. When x = 1 this function is not defined,                                              lim [f (x) + g(x)] = L1 + L2
                                                                                                                                      x→a
it is zero over zero. Hence                                                                                                           lim [f (x) − g(x)] = L1 − L2
                                                                                                                                      x→a
                                                  x−1      x=1                                                                           lim [f (x)g(x)] = L1 L2                                (9)
                                     f (x) =                                                  (4)                                        x→a
                                                  undefined x = 1
                                                                                                    and, provided L2 = 0
However, the limit at x = 1 does not depend on the value at x = 1, so                                                                                f (x)   L1
                                                                                                                                               lim         =                                   (10)
                                                                                                                                             x→a     g(x)    L2
                                 2                   2
                                x −x       x − x x(x − 1)                                           If L1 > 0
                          lim        = lim                lim x = 1                           (5)
                          x→1   x−1    x→1 x − 1  x − 1 x→1
                                                                                                                                            lim      f (x) =   L1                              (11)
                                                                                                                                            x→a
Now, although this looks like a somewhat contrived example, it is not in fact so unusual,           In fact, once we know that continuity is, this last result is a special case of a more general
in definition of the derivative                                                                      theorem: if lim x → af (x) = L1 and g(x) is continuous at x = L1 then
                                      df       f (x + h) − f (x)                                                                            lim g(f (x)) = g(L1 )                              (12)
                                         = lim                                                (6)                                           x→a
                                      dx h→0           h
                                                                                                                    √                                                 √
  1
      Conor Houghton, houghton@maths.tcd.ie, see also http://www.maths.tcd.ie/~houghton/231         So, if g(x) =       x then this reduces to limx→a       f (x) =       L1 .

                                                         1                                                                                              2
The one-sided limit                                                                           we have illustrated above, if x < 4 then the closer x gets to 4, the closer the function gets
                                                                                              to 4, however, if x > 4 then the closer x gets to 4, the closer f (x) gets to 3. Hence
There are lots of reason why a limit might not exist: one obvious one is that there is a
discontinuity. Consider                                                                                                                 lim f (x) = 4
                                                                                                                                    x→4−
                                         x      x<4                                                                                     lim f (x) = 3                                 (17)
                                f (x) =                                            (13)                                             x→4+
                                         7−x x>4
                                                                                              However, there is no limit as x → 4, just specifying that x is close to 4 doesn’t tell you
                                                                                              that f (x) is close to a particular value, precisely because f (x) could be close to 4 or 3
                                                                                              depending on whether x is bigger or smaller than 4.
                        6
                                                                                                 The formal definition is very similar to the formal definition for the limit, now, for the
                        5                                                                     limit from below we specify that x < a and, for the limit from above, that x > a.
                                                                                              One-sided Limit: (Formal definition)
                        4
                                                                                                                                        lim f (x) = L                                 (18)
                                                                                                                                        x→a+
                        3
                                                                                              if and only if for all ǫ > 0 there exists a δ > 0 such that for all x > a such that x − a < δ
                        2                                                                     |f (x) − L| < ǫ and
                                                                                                                                        lim f (x) = L                                  (19)
                                                                                                                                        x→a−
                        1
                                                                                              if and only if for all ǫ > 0 there exists a δ > 0 such that for all x < a such that a − x < δ
                                                                                              |f (x) − L| < ǫ.
                                1    2      3        4   5   6   7
                                                                                              Infinite limits
                                                                                              Another reason a limit, or even a one-sided limit might not exist is that the function
There is a discontinuity at x = 4 and so there the limit                                      approaches an infinitely large negative or positive number, that is, at x = a the function
                                                                                              is undefined because it has the form 1/0. Obvious examples would be f (x) = 1/x
                                           lim f (x)                                  (14)
                                           x→4

does not exist. It is nonetheless useful to have a mathematical language to describe this
situation, that language is the one-sided limit.                                                                                               4

One-sided Limit: (Informal definition). If the value of f (x) can be made as close as we
like to L by taking values of x sufficiently close to a and greater than a then we write
                                                                                                                                               2

                                         lim f (x) = L                                (15)
                                      x→a+

If the value of f (x) can be made as close as we like to L by taking values of x sufficiently                                                    0
                                                                                                                          -4       -2               0   2    4
close to a and less than a then we write

                                         lim f (x) = L                                (16)                                                     -2
                                      x→a−

   Thus, for the limit from below, x → a−, we are only consider where the function is
going as x approaches a from below, conversely, for the limit from above, x → a+, we are                                                       -4

only considering where the function is going as x approaches a from above. In the example


                                                 3                                                                                                  4
and f (x) = 1/x2                                                                                                                                1
                                                                                                                                         lim      = ∞                                      (24)
                                                                                                                                        x→0+    x
                                                  10                                                  The formal definition has to find a mathematical language for the English ‘as large as
                                                                                                   we like’:
                                                                                                   Infinite Limit: (Formal definition).
                                                  8

                                                                                                                                          lim f (x) = ∞                                    (25)
                                                                                                                                          x→a

                                                  6
                                                                                                   if and only if for all M > 0 there exists a δ > 0 such that for all x such that |x − a| < δ
                                                                                                   f (x) > M
                                                                                                       It does that by demanding that for all values of M, f (x) can be made bigger than M
                                                  4
                                                                                                   by taking x close enough to a. There are similar definitions for the one-sided limits, so, for
                                                                                                   example, example
                                                  2
                                                                                                   Infinite one-sided limit from above: (Formal definition).

                                                                                                                                          lim f (x) = ∞                                    (26)
                                                                                                                                         x→a+
                                                  0
                              -4      -2               0   2     4                                 if and only if for all M > 0 there exists a δ > 0 such that for all x > a such that x − a < δ
According to our previous definitions, these functions have no limit at x = 0, there is             f (x) > M
no number they approach as x approaches zero. However, it is useful to deal with these
examples by extending the definition of the limit to include infinite limits.
Infinite Limit: (Informal definition). If the value of f (x) can be made as large as we like
by taking values of x sufficiently close to a but not equal to a then we write
                                           lim f (x) = ∞                                   (20)
                                           x→a

if the value can be made as large a negative number as we like by taking values of x
sufficiently close to a but not equal to it, we write
                                           lim f (x) = ∞                                   (21)
                                           x→a

There are also one-sided versions of this, for example If the value of f (x) can be made as
large as we like by taking values of x sufficiently close to a and greater than a then we
write
                                       lim f (x) = ∞                                   (22)
                                           x→a+

   Now, returning to f (x) = 1/x2 , the closer x gets to zero, the larger 1/x2 become, so
                                                  1
                                            lim      =∞                                    (23)
                                            x→0   x2
In contrast, 1/x is negative if x is negative and positive as x is positive, but the denominator
gets smaller and smaller as x gets smaller, while the numerator does not, so
                                                 1
                                       lim         = −∞
                                       x→0−      x

                                                       5                                                                                         6

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:12
posted:11/27/2011
language:English
pages:3