University Mathematics Handbook (2015)

IV. Single-Variable Differential Calculus

Chapter 2. Limits of Functions

2.1  Definitions of Limit

a.  Heine's Definition of Limit

1.  Let function be defined in the neighborhood of point , except, possibly, point itself. A real number is the limit of when if, for all sequence converging to limit , such that , the sequence converges to .

2.  Let function , be defined in ray . A real number is the limit of when tends to infinity, if for all sequence converging to infinity, the sequence converges to .

b.  Cauchy's Definition of Limit

1.  Let function be defined in the neighborhood of point , except, possibly, point itself. is the limit of when , , if for every arbitrary small number there exists a such that for all which holds , there holds , or, in logic symbols

.

2.   is the limit of when tends to infinity, if, for an arbitrary small there exists such that for all holding , there holds or, in logic symbols

.

c.  The limit of at point , is infinity , if for all there exists such that for all that holds , there holds .

d.  Heine's and Cauchy's definitions are equivalent.

2.2  Properties of Limits

a.  Let and be two functions, and , are their limits. Then:

1.  , for every constant

2.  

3.  

4.  

5.  If , then

b.  If function is bounded and , then .

Example: and .

Therefore, .

c.  Squeeze (sandwich) theorem: Let , , be three functions defined in a neighborhood of point , except, possibly, point itself. If, for all in this neighborhood, there holds and the limits exists, then, .

d.  If function has a limit, then it is unique.

e.  If there exist two different sequences , converging to , but sequences and converges to different limit, then, there is no limit for when .

Example: For function , when is the integer value of , there exists no limit when .

Proof: Let's take the sequence , holding , and, for all , . Therefore, , and therefore . Now, let's take the sequence . Here, . For all , , and . That is, there doesn't exist a limit .

f.  If is an elementary function, (see, I, 1.9) defined at point , then .

g.  .

h.  , .

i.  If function is defined in the neighborhood of and there exists a limit , then:

1.  There exists a neighborhood of where is bounded.

2.  If , then there exists a neighborhood of , such that for all in this neighborhood, (except, possibly, ), where there holds .

3.  If , then there exists a neighborhood of , such that for all in this neighborhood, (except, possibly, ), where there holds .

2.3  One-Sided Limits

a.  Let be a function defined in the right-handed neighborhood of point . That is, let there exist such that is defined in interval . Real number is the right-handed limit of when tends to point to the right , (always ), if, for every , there exists such that

.

Let's write .

b.  Let be a function defined in the left-handed neighborhood of point . That is, let there exist such that is defined in interval . Real number is the left-handed limit of when , if for every there exists such that holds . Let's write .

Example: , .

c.  For function , there exists a limit at point if, and only if, there exist one-sided limits at this point, and holds.