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| 0, we can make |f (x) − L| 0 such that if |x − a| 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 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 x4
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.
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 0 there exists a δ > 0 such that for all x 0 there exists a δ > 0 such that for all x such that |x − a| 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 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
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