Let be a function and let . We say that has a limit as approaches , or that is the limit of at , if for any positive tolerance , we can find a cutoff distance such that if the distance from to is less than , and if , then approximates with an error less than .

That is, if , then .

In this case, we write

(Note: We sometimes write as shorthand for "x approaches a" and as shorthand for ".")

Given a function that is defined on an open interval containing , but potentially excluding the point , we have the following equivalence related to limits:

1. The limit of as approaches exists and equals .
2. If is a sequence such that and as , then .

To understand why limits work, suppose . If we have a sequence with and , then for large , should be close to .

Given any , if , there exists so that if is within of (excluding ), is within of .

If , for sufficiently large (say ), is within of . This implies is within of , confirming our belief that .

This shows that 1→2 but we want to show that 2→1 as well. We will show that if is not the limit of then 2 fails.

Consider the scenario where we can define a sequence such that and , yet the sequence doesn't converge to .

Assuming that is not the limit of as approaches , there must be an such that for every and for any satisfying , the inequality

doesn't hold. Hence, for every , there exists a in the interval (excluding ) such that

Given any natural number , if we consider , then there is an with but still, the inequality

is satisfied.

This gives us a sequence such that and . However, for each , the following condition holds:

From this, it's evident that the sequence does not converge to .

Assignment 2.pdf > page=21

Forrest_M147CN_F20 (1).pdf > page=173

Hmmm? #TA

Let and be functions and let . Assume that
and that
Then:

i) Assume that for every . Then

ii) For any ,

iii)

iv)

v)

vi)

Proof:

Take

By Sequential Characterization

We can use these to apply the Arithmetic Properties of Limits of Sequences, for eg.

Note: We can Do Something Similar for Squeeze Theorem Squeeze Theorem

Real Analysis Theorems > Theorem 3

Let be a function and let .

We say that has a limit as approaches from the right, or from above, if for any positive tolerance , we can find a cutoff distance such that if the distance from to is less than , and if , then approximates with an error less than . That is, if , then .

In this case, we write

Note that the positive superscript, “+”, indicates that this is a limit from the right.

Let be a function and let .

We say that has a limit as approaches from the left, or from below, if for any positive tolerance , we can find a cutoff distance such that if the distance from to is less than , and if , then approximates with an error less than . That is, if , then .

In this case, we write

Note that the positive superscript, “-”, indicates that this is a limit from the left.

Let be a function defined on an open interval containing except possibly at . Then the following two statements are logically equivalent:

1. exists and equals .

2. Both one-sided limits exist, and
   .