Precise definition of limits The phrases “x is close to a” and “f(x) gets closer and closer to L” are vague. since f(x) can be arbitrarily close to 5 as long as x approaches 3 sufficiently. How close to 3 does x have to be so that f(x) differs from 5 by less than 0.1? Solving the inequality |(2x-1)-5|<0.1, we get |x-3|<0.05, i.e., we find a number =0.05 such that whenever |x-3|<we have |f(x)-5|<0.1
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Precise definition of limits The phrases “x is close to a” and “f(x) gets closer and closer to L” are vague. since f(x) can be arbitrarily close to 5 as.
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Precise definition of limits The phrases “x is close to a” and “f(x) gets closer
and closer to L” are vague. since f(x) can be arbitrarily close to 5 as long as x approaches 3 sufficiently. How close to 3 does x have to be so that f(x)
differs from 5 by less than 0.1? Solving the inequality |(2x-1)-5|<0.1, we get |x-3|<0.05, i.e., we find a number =0.05 such that whenever |x-3|< we have |f(x)-5|<0.1
definition of a limit If we change the number 0.1 to other smaller
numbers, we can find other s. Changing 0.1 to
any positive real number , we have the following Definition: We say that the limit of f(x) as x
approaches a is L, and we write if
for any number >0 there is a number >0 such that
Remark expresses “arbitrarily” and expresses
“sufficiently” Generally depends on To prove a limit, finding is the key point means that for every >0 (no matter
how small is) we can find >0 such that if x lies in
the open interval (a-,a+) and xa then f(x) lies in
the open interval (L-,L+).
Example
Ex. Prove that
Sol. We solve the question in two steps.
1. Preliminary analysis of the problem (deriving a
value for ). Let be a given positive number, we
want to find a number such that
But |(4x-5)-7|=|4x-12|=4|x-3|, therefore we want
Example (cont.)
This suggests that we should choose =/.
2. Proof (showing the above works). Given choose If 0<|x-3|<, then
|(4x-5)-7|=|4x-12|=4|x-3|<4Thus
Therefore, by definition we have
Example
Ex. Prove that
Sol. 1. Deriving a value for . Let >0 be given, we
want to find a number such that
Since |(x2-x+2)-4|=|x-2||x+1|,if we can find a positive
constant C such that |x+1|<C, then |x-2||x+1|<C|x-2|
and we can make C|x-2|< by taking |x-2|</CAs we
are only interested in values of x that are close to 2,
Example (cont.)
it is reasonable to assume |x-2|<1. Then 1<x<3, so
2<x+1<4, and |x+1|<4. Thus we can choose C=4 for
the constant. But note that we have two restrictions on
|x-2|, namely, |x-2|<1 and |x-2|</C=/4. To make sure
both of the two inequalities are satisfied, we take to
be the smaller of 1 and /4. The notation for this is
=min{1,/4}.
2. Showing above works. Given >0, let =min{1,/4}.
Example (cont.)
If 0<|x-2|<, then |x-2|<1) 1<x<3) |x+1|<4. We also
have |x-2|</4, so |(x2-x+2)-4|=|x-2||x+1|</4¢4=This
shows that
can be found by solving the inequality, but no need to solve the inequality: is not unique, finding one is enough
Example
Ex. Prove that
Sol. For any given >0, we want to find a number >0
such that
By rationalization of numerator,
If we first restrict x to |x-4|<1, then 3<x<5 and
Example (cont.)
Now we have and we can make
by taking Therefore
If >0 is given, let
When 0<|x-4|<we have firstly
and then
This completes the proof.
Proof of uniqueness of limits(uniqueness) If and then K=L.
Proof. Let >0 be given, there is a number 1>0 such that
|f(x)-K|< whenever 0<|x-a|<1. On the other hand, there is
a number 2>0 such that |f(x)-L|<whenever 0<|x-a|<2.
Now put =min{1,2} and x0=a+Then|f(x0)-K|<
and |f(x0)-L|<. Thus |K-L|=|(f(x0)-K)-(f(x0)-L)|·|f(x0)-K|+
|f(x0)-L|<2. Since is arbitrary, |K-L|<2 implies K=L.
definition of one-sided limits
Definition: If for any number >0 there is a number
>0 such that
then
Definition: If for any number >0 there is a number
>0 such that
then
Useful notations 9 means “there exist”, 8 means “for any”. definition using notations
such that
there holds
,
0, 0, : 0 | | ,x x a
| ( ) | .f x L
M- definition of infinite limits
Definition. means that
8 M>0, 9 >0, such that
whenever
Remark. M represents “arbitrarily large”
( )f x M 0 | | .x a
Negative infinity means
Continuity Definition A function f is continuous at a number a if
Remark The continuity of f at a requires three things:
1. f(a) is defined
2. The limit exists
3. The limit equals f(a)
otherwise, we say f is discontinuous at a.
).()(lim afxfax
)(lim xfax
)(lim xfax
Continuity of essential functionsTheorem The following types of functions are continuous