I am learning calculus now I want to type up everything I now about this subject. All solutions written below, if no source is mentioned, are my solutions so if someone find an error in the solutions, please tell me. By the way, thanks to Luca Trevisan’s LaTeX to WordPress, I now don’t have to write LaTeX in WordPress format anymore, which is very convenient.
1.1. Definition of limits of function
For me, the precise definition of limits is a bit hard to understand so I decided to reconstruct the whole thing by writing some of my guesses on why did mathematicians write the definition in this way. By the way, French mathematician, Augustin-Louis Cauchy (1789-1857) is the one who introduced into definition of limits.
Limit, the notation is understood vaguely as: as gets closer to , gets closer to a number . What does it mean by ‘get closer’? It is easier to understand if we imagine lies on a number line and suppose we pick an arbitrary point on the line. If we apply the term ‘ gets closer to ‘, we can understand it as distance from to on the number line decreases. Its distance is so saying gets closer to is the same as saying gets closer to . Note that for any distance , there are two possible on the number line that can satisfy that distance. What we care about is the distance from to not the position of , so using ‘ gets closer to ‘ is better than ‘ gets closer to ‘. Similarly to and we get a new definition:
Note that we can get as close to as we want by choosing appropriate , but we can’t guarantee the same thing for since depends on . Thus, firsly, in order for to be able to approach we need to be as small as we want. It is equivalent to saying:
For any there exists some so .
Secondly, we need the definition to say approaches as approaches . This means for small enough , must be bounded by some for all . For better understanding more about this argument, let’s assume the contrary, then as gets closer to , still does not have a bound for all . Say we find at has . We consider all in and follow that there must be an so and according to the assumption. Next consider all in and from our assumption, there exists so for . This keeps going and we can see that as gets closer to , gets larger, which contradicts to our aim that must approaches . Thus, we got the second part of the definition:
For small enough , must be bounded by a number for all .
According to first argument, the second argument has to be true for any and note that in second argument, depends on choice of . Thus, by combining the two we have:
For any , there must exists a so for all .
Thus, we obtain the following definition:
To be more clearly, in order for to be able to approach while is still defined then must be defined on some open interval that contains :
According to Definition 3 then the limit of function:
at will not be defined. Why? Because we can’t find a limit of as goes to . From the graph, we find that except at , all points approach (or approaches ) as approaches , i.e. for any there exists so if then . But since definition 3 includes point at and from the graph, we can clearly see that at , does not approach . In other words, at then so we can’t choose to be true for all . So the limit is definitely not . It also can’t be because other points () won’t agree with it. So all we can conclude from definition 3 is that function doesn’t have limit at . Mathematicians wanted functions such as to have limit at so they decided to take out the value of function at from definition of limit:
So with this, limit of at is . That means we don’t actually care about what happen at , we only care about what happen near .
From this definition, we can see that the smaller we choose , the smaller . This is equivalent to saying that distance between and can be made arbitrarily small if we take distance from to sufficiently small. This agrees with our intuition about limits.
1.2. More definitions of limits
In above definition, can approach from two sides: left of and right of . We can define limit analogous if only approach from only one side, simply by changing
Similarly for right-hand limit . Next, what if apprroaches infinity as approaches ?
We can define similarly for . What about limit as goes to infinity.
We can also define convergence of sequences in a similar way:
1.3. Another way to define limits of function
We can rewrite the definition 3 in terms of limits of sequence.
Proof: (3) (9): Consider sequence which converges to . We want to prove that sequence converges to i.e. for any , there exists so that for all then . Indeed, from definition 3, for any , there exists so for all then . But since converges so for such , there exists so that for all then . Thus, for each , there exists so for all , and for that there exists so for all . This follows for each , there exists so for all .
(9) (3): It suffices to prove that for any there is so for all . Indeed, consider all sequences which consists entirely elements of and converges to then from definition 9, sequences converges to . Hence, for each sequence there exists a so for all then . Now we need to use prove existence of so that this is true for all , i.e. for each there exists a so
- for each .
- to ensure that at the same time as for all sequences .
- Each in belongs to at least one of the sequence and equals to with . In other words, is a subset of union of all terms in all sequences which converge to .
Condition 1 can be done according to definition 8 (convergence of sequences). From definition 8, we also follow that if we pick smaller and smaller then must get bigger and bigger. Hence, condition 2 can be satisfied. Condition 3, assume a contrary that now matter how small we choose, we still can’t cover with . This means for each , there still exists a sequence which has with but , which means . For each , we take out such . By letting smaller and smaller, we’ve create a sequence which converges to (true according to definition 8) so that . This contradicts to our condition of definition 9. Thus, we follow that condition 3 will eventually be true if we take sufficiently small. Hence, all three conditions can be obtain if we take sufficiently small. And for such then is true for all . We are done.
Are definitions 4 and 9 equivalent to each other? Logically, there is a possibility no matter how large is, there always exists so . And if we pick so is really far away from , then cannot converge to . Definition 4 ignores value of at but definition 8 (limits of sequences) does include value of at , which follows two things:
- can be equal to for infinitely many .
- If we want to prove converges to as converges to , then we also need to show for every (this we assume that for infinitely many ).
The second claim can’t be true if we pick such so is far away from . Hence, we can’t claim converges or diverges if we don’t know any further information of . In conclusion, the answer is no, definitions 4 and 9 are not equivalent to each other.
We can take function in (1) as an example to support above argument. According to definition 4, . Consider arbitrary sequence of that converges to , and then among them, say add more for infinitely many . The new sequence still converges to , but we see that does not converge to since there are infinitely many so .
1.4. Uniqueness of limits of functions
Proof: If and then that means for all there exists so for all and for all . Pick then for all we have and . Therefore,
If we can pick small enough so , which leads to a contradiction. Thus, .
Uniqueness of other limits (left-hand, infinite limits, sequences) can be proved similarly to above.