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Limits

1. Limits in general

If f is some function then:

(lim_(x→a)) f(x) = L 

is read “the limit of f(x) as x approaches a is L”. It means that if you choose values of x which are close but not equal to a, then f(x) will be closed to the value L; moreover f(x) gets closer and closer to L as x gets closer and closer to a.

The following alternative notation is sometimes ussed 
f(x) → L  as   x → a.

where a is a free variable and x is an adummy variable(“placeholder” or bound variable).


Definition 1.1.
We say that L is the limit of f(x) as x → a, if:

(1) f(x) need not to defined as x = a, but it must be defined for all other x in some interval which contains a.
(2) for every ϵ > 0 there is a δ > 0 such that for all x in the domain of f one has:
| x - a | < δ  implies | f(x) - L | < ϵ .

Where inequlity | x - a | < δ say that “the distance between x and a is less than δ ”.

2. Left and right hand limits

When we let “x approach a” we allow x to be both larger or smaller than a, as long as x gets close to a. If we explicitly want to study the behaviout of f(x) as x approaches a through vales larger than a, then we wrtire:

(lim_(x↘a)) f(x)    or   (lim_(x→(SuperPlus[a]))) f(x)    or    (lim_(x→(SuperPlus[a])0)) f(x)    or    (lim_(x→a,x>a)) f(x) 

All four notations are in use. Similarly, to designate the value which f(x) approaches as x approaches a through values below a one writes:

(lim_(x↗a)) f(x)    or   (lim_(x→(SuperMinus[a]))) f(x)    or    (lim_(x→(SuperMinus[a])0)) f(x)    or    (lim_(x→a,x<a)) f(x) 


Definition 2.1. Right-limits. Let f be a function. Then  (lim_(x→(SuperPlus[a]))) f(x)  means that for every  ϵ > 0 one can find a δ > 0 such that:

a < x < a+δ  ⟹  | f(x) - L | < ϵ

holds for all x in the domain of f.


Definition 2.2. Left-limits. Let f be a function. Then  (lim_(x→(SuperMinus[a]))) f(x)  means that for every  ϵ > 0 one can find a δ > 0 such that:

a - δ  < x < a   ⟹  | f(x) - L | < ϵ

holds for all x in the domain of f.

Theorem 2.1.
If both one-sided limits (lim_(x→(SuperPlus[a]))) f(x) = (SubPlus[L]) and (lim_(x→(SuperMinus[a]))) f(x) = (Subscript[L,  -]) exist, then
(lim_(x→a)) f(x) exists ⟺  (SubPlus[L]) = (Subscript[L,  -]).

In other words, if function has both left- and right- limits at same x=a, then that function has a limit at x=a if the left- and right- limits are equal.

3. Limits at infinity

Instead of letting x approach some finite number, one can letx become “larger and larger” and ask what happens to f(x). If there is a number L such that f(x) gets arbitrality close to L if one chooses x sufficiently larger, then we write

(lim_(x→∞)) f(x) = L,    or   (lim_(x↑∞)) f(x) = L,   or    (lim_(x↗∞)) f(x) = L. 

(“The limit for x going to infinity is L.”)

Definition 3.1. 
Let f be some function which is defined on some interval (Subscript[x, 0]) < x < ∞. If there is a number L such that for every ϵ > 0 one can find an A such that:

x > A  ⟹  | f(x) - L | < ϵ

for all x, then we say that limit of f(x) for x ⟶ ∞  is L.

4. Properties of the limits

4.1 Limits of constants and of x

If a and c are constants then
(lim_(x→a)) c = c   and   (lim_(x→a)) x = a
(lim_(x→a)) [c f(x)] = c (lim_(x→a))  f(x)

4.2. Limits of sums, products and quotients

Let (Subscript[f, 1])  and (Subscript[f, 2])  be two given functions whose limits for x → a we know:  (lim_(x→a)) (Subscript[f, 1])(x) = (Subscript[L, 1]) and (lim_(x→a)) (Subscript[f, 2])(x) = (Subscript[L, 2]) then:

(lim_(x→a)) ( (Subscript[f, 1])(x) +  (Subscript[f, 2])(x) ) = (Subscript[L, 1]) + (Subscript[L, 2])
(lim_(x→a)) ( (Subscript[f, 1])(x) -  (Subscript[f, 2])(x) ) = (Subscript[L, 1]) - (Subscript[L, 2])
(lim_(x→a)) ( (Subscript[f, 1])(x)  ·  (Subscript[f, 2])(x) ) = (Subscript[L, 1]) · (Subscript[L, 2])

Furthermore if  (lim_(x→a)) (Subscript[f, 2])(x) ≠ 0, 

(lim_(x→a))  (((Subscript[f, 1])(x) )/((Subscript[f, 2])(x)))  =  ((Subscript[L, 1])/(Subscript[L, 2]))

5. When limits fail to exist

Definition 5.1.
If there is no number L such that (lim_(x→a)) f(x) = L, then we say that the limit (lim_(x→a)) f(x)  does not exist.

6. Inequlities in limits

Theorem 6.1.
 Let f and g be functions whose limits for x→a exist, and assume that f(x) ⩽ g(x) holds for all x. Then
(lim_(x→a)) f(x) ⩽ (lim_(x→a)) g(x)


Theorem 6.2.
The Sandwich Theorem: Suppose that  f(x) ⩽ g(x) ⩽ h(x) for all x and that (lim_(x→a)) f(x) = (lim_(x→a)) h(x) then 
(lim_(x→a)) f(x) =  (lim_(x→a)) g(x) = (lim_(x→a)) h(x)

7. Limits and continuity of function

Definition 7.1. A function g is continuous at a if 
 (lim_(x→a)) g(x) = g(a)
A function is continuous if it is continuous at every a in its domain.

8. Substitution in limits

Theorem 8.1.
If   (lim_(x→a)) g(x) = a, and if the function f is continuous at u =L, then 

(lim_(x→a)) f(g(x)) =  (lim_(u→L)) f(u) = f(L).

Another way to write this is 

(lim_(x→a)) f(g(x)) = f( (lim_(x→a)) g(x) ).

9. Important examples of limits

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10. L’Hospital Rule

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