A limit
on the left (a left-hand limit) and a limit
on the right (a right-hand limit) |
Continuous
function |
Limits
at infinity (or limits of functions as x approaches
positive or negative infinity) |
Infinite
limits |
The
limit of a function examples |
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The
definition of the limit of a function |
The
limit of a function is a real number L
that f
(x)
approaches as x
approaches a given real number a,
written |
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if
for any e
> 0 there is a d(e)
> 0 such that |
f
(x)
-
L | < e
whenever | x
-
a | < d(e). |
The
definition says, no matter how small a positive |
number
e
we take, we can find a positive number |
d
such that, for an arbitrary chosen value of x
from |
the interval a
-
d
< x
< a +
d, |
the corresponding function's values lie inside |
the interval
L
-
e
< f(x)
< L +
e, |
as shows the right figure. |
That
is, the function's values can be made arbitrarily |
close to the number L
by choosing x
sufficiently
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close to
a, but not equal to
a. |
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Therefore,
the number d,
that measures the distance between a point x
from the point a
on the x-axis,
depends
on the number e
that measures the distance between the point f
(x)
from the point L
on the y-axis. |
Example: Given |
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whenever |
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A
limit is used to examine the behavior of a function near a point
but not at the point. The function need not even be defined at
the point. |
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A
limit on the left (a left-hand limit) and a limit
on the right (a right-hand limit) |
The
limit of a function where the variable x
approaches the point a
from the left or, where x
is restricted to values
less than a,
is written |
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The
limit of a function where the variable x
approaches the point a
from the right or, where x
is restricted to values
grater than a,
is written |
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If
a function has both a left-handed limit and a right-handed limit
and they are equal, then it has a limit at the point.
Thus,
if |
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Continuous
function |
A
real function
y = f (x)
is continuous at a point a
if it is defined at x =
a
and |
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that
is, if for every e
> 0 there is a d(e)
> 0 such that |
f
(x)
-
f
(a)
| < e
whenever | x
-
a | < d(e). |
Therefore,
if a function changes gradually as independent variable changes,
so that at every value a,
of the independent
variable, the difference between f
(x)
and f
(a)
approaches zero as x
approaches a. |
A
function is said to be continuous if it is continuous at all
points. |
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Limits
at infinity (or limits of functions as x approaches
positive or negative infinity) |
We
say that the limit of f
(x)
as x
approaches positive infinity is L
and write, |
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if for any e
> 0 there exists N
> 0 such that
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f
(x)
-
L | < e
for all x
> N(e). |
We
say that the limit of f
(x)
as x
approaches negative infinity is L
and write, |
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if for any e
> 0 there exists N
> 0 such that
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f
(x)
-
L | < e
for all x
< -N(e). |
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Not
all functions have real limits as x
tends to plus or minus infinity. |
Thus
for example, if f
(x)
tends to infinity as x
tends to infinity we write |
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if
for every number N
> 0 there is a number M
> 0 such that f
(x)
> N whenever
x
> M(N). |
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Infinite limits |
We
write |
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if
f (x) can
be made arbitrarily large by choosing x
sufficiently close
but not equal to a. |
We
write |
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if
f (x) can
be made arbitrarily large negative by choosing x
sufficiently close
but not equal to a. |
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The
limit of a function examples |
Example: Evaluate
the following limits; |
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Solution:
a) As x
tends to minus infinity f
(x) |
gets closer and closer to 0. |
As x
tends to plus infinity f
(x)
gets closer and |
closer to 0. Therefore, |
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b)
As x
tends to 0 from the left f
(x)
gets larger in |
negative sense. |
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As
x
tends to 0 from the right f
(x)
gets larger in positive sense. Therefore, |
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