Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Analysis of Functions of a Single Variable » The Limit of a Function

Navigation

Table of Contents

Recently Viewed

This feature requires Javascript to be enabled.
 

The Limit of a Function

Module by: Lawrence Baggett. E-mail the author

Summary: The concept of the derivative of a function is what most people think of as the beginning of calculus. However, before we can even define the derivative we must introduce a kind of generalization of the notion of continuity. That is, we must begin with the definition of the limit of a function.

The concept of the derivative of a function is what most people think of as the beginning of calculus. However, before we can even define the derivative we must introduce a kind of generalization of the notion of continuity. That is, we must begin with the definition of the limit of a function.

Definition 1:

Let f:S→Cf:S→C be a function, where S⊆C,S⊆C, and let cc be a limit point of SS that is not necessarily an element of S.S. We say that ffhas a limit L as z approaches c, and we write

lim z → c f ( z ) = L , lim z → c f ( z ) = L ,
(1)

if for every ϵ>0ϵ>0 there exists a δ>0δ>0 such that if z∈Sz∈S and 0<|z-c|<δ,0<|z-c|<δ, then |f(z)-L|<ϵ.|f(z)-L|<ϵ.

If the domain SS is unbounded, we say that f has a limit L as z approaches∞,∞, and we write

L = lim z → ∞ f ( z ) , L = lim z → ∞ f ( z ) ,
(2)

if for every ϵ>0ϵ>0 there exists a positive number BB such that if z∈Sz∈S and |z|≥B,|z|≥B, then |f(z)-L|<ϵ.|f(z)-L|<ϵ.

Analogously, if S⊆R,S⊆R, we say limx→∞f(x)=Llimx→∞f(x)=L if for every ϵ>0ϵ>0 there exists a real number BB such that if x∈Sx∈S and x≥B,x≥B, then |f(x)-L|<ϵ.|f(x)-L|<ϵ. And we say that limx→-∞f(x)=Llimx→-∞f(x)=L if for every ϵ>0ϵ>0 there exists a real number BB such that if x∈Sx∈S and x≤B,x≤B, then |f(x)-L|<ϵ.|f(x)-L|<ϵ.

Finally, for f:(a,b)→Cf:(a,b)→C a function of a real variable, and for c∈[a,b],c∈[a,b], we define the one-sided (left and right) limits of ff at c.c. We say that ff has a left hand limit of LL at c,c, and we write L=limx→c-0f(x),L=limx→c-0f(x), if for every ϵ>0ϵ>0 there exists a δ>0δ>0 such that if x∈(a,b)x∈(a,b) and 0<c-x<δ0<c-x<δ then |f(x)-L|<ϵ.|f(x)-L|<ϵ. We say that ff has a right hand limit of LL at c,c, and write L=limx→c+0f(x),L=limx→c+0f(x), if for every ϵ>0ϵ>0 there exists a δ>0δ>0 such that if x∈Sx∈S and 0<x-c<δ0<x-c<δ then |f(x)-L|<ϵ.|f(x)-L|<ϵ.

The first few results about limits of functions are not surprising. The analogy between functions having limits and functions being continuous is very close, so that for every elementary result about continuous functions there will be a companion result about limits of functions.

Theorem 1

Let cc be a complex number. Let f:S→Cf:S→C and g:S→Cg:S→C be functions. Assume that both ffand gg have limits as xx approaches c.c. Then:

  1. There exists a δ>0δ>0 and a positive number MM such that if z∈Sz∈S and 0<|z-c|<δ0<|z-c|<δ then |f(z)|<M.|f(z)|<M. That is, if ff has a limit as zz approaches c,c, then ff is bounded near c.c.
  2. lim z → c ( f ( z ) + g ( z ) ) = lim z → c f ( z ) + lim z → c g ( z ) . lim z → c ( f ( z ) + g ( z ) ) = lim z → c f ( z ) + lim z → c g ( z ) .
    (3)
  3. lim z → c ( f ( z ) g ( z ) ) = lim z → c f ( z ) lim z → c g ( z ) . lim z → c ( f ( z ) g ( z ) ) = lim z → c f ( z ) lim z → c g ( z ) .
    (4)
  4. If limz→cg(z)≠0,limz→cg(z)≠0, then
    limz→cf(z)g(z)=limz→cf(z)limz→cg(z),limz→cf(z)g(z)=limz→cf(z)limz→cg(z),
    (5)
  5. If uu and vv are the real and imaginary parts of a complex-valued function f,f, then uu and vv have limits as zz approaches cc if and only if ff has a limit as zz approaches c.c. And,
    limz→cf(z)=limz→cu(z)+ilimz→cv(z).limz→cf(z)=limz→cu(z)+ilimz→cv(z).
    (6)

Exercise 1

  1. Prove Theorem 1. HINT: Compare with (Reference).
  2. Prove that limx→cf(x)=Llimx→cf(x)=L if and only if, for every sequence {xn}{xn} of elements of SS that converges to c,c, we have limf(xn)=L.limf(xn)=L. HINT: Compare with (Reference).
  3. Prove the analog of Theorem 1 replacing the limit as zz approaches cc by the limit as zz approaches ∞.∞.

Exercise 2

  1. Prove that a function f:S→Cf:S→C is continuous at a point cc of SS if and only if limx→cf(x)=f(c).limx→cf(x)=f(c). HINT: Carefully write down both definitions, and observe that they are verbetim the same.
  2. Let ff be a function with domain S,S, and let cc be a limit point of SS that is not in S.S. Suppose gg is a function with domain S∪{c},S∪{c}, that f(x)=g(x)f(x)=g(x) for all x∈S,x∈S, and that gg is continuous at c.c. Prove that limx→cf(x)=g(c).limx→cf(x)=g(c).

Exercise 3

Prove that the following functions ff have the specified limits LL at the given points c.c.

  1. f(x)=(x3-8)/(x2-4),c=2,f(x)=(x3-8)/(x2-4),c=2, and L=3.L=3.
  2. f(x)=(x2+1)/(x3+1),c=1,f(x)=(x2+1)/(x3+1),c=1, and L=1.L=1.
  3. f(x)=(x8-1)/(x6+1),c=i,f(x)=(x8-1)/(x6+1),c=i, and L=-4/3.L=-4/3.
  4. f(x)=(sin(x)+cos(x)-exp(x))/(x2),c=0,f(x)=(sin(x)+cos(x)-exp(x))/(x2),c=0, and L=-1.L=-1.

Exercise 4

Define ff on the set SS of all nonzero real numbers by f(x)=cf(x)=c if x<0x<0 and f(x)=df(x)=d if x>0.x>0. Show that limx→0f(x)limx→0f(x) exists if and only if c=d.c=d.

(b) Let f:(a,b)→Cf:(a,b)→C be a complex-valued function on the open interval (a,b).(a,b). Suppose cc is a point of (a,b).(a,b). Prove that limx→cf(x)limx→cf(x) exists if and only if the two one-sided limits limx→c-0f(x)limx→c-0f(x) and limx→c+0f(x)limx→c+0f(x) exist and are equal.

Exercise 5: Change of variable in a limit

Suppose f:S→Cf:S→C is a function, and that limx→cf(x)=L.limx→cf(x)=L. Define a function gg by g(y)=f(y+c).g(y)=f(y+c).

  1. What is the domain of g?g?
  2. Show that 0 is a limit point of the domain of gg and that limy→0g(y)=limx→cf(x).limy→0g(y)=limx→cf(x).
  3. Suppose T⊆C,T⊆C, that h:T→S,h:T→S, and that limy→dh(y)=c.limy→dh(y)=c. Prove that
    limy→df(h(y))=limx→cf(x)=L.limy→df(h(y))=limx→cf(x)=L.
    (7)

REMARK When we use the word “ interior” in connection with a set S,S, it is obviously important to understand the context; i.e., is SS being thought of as a set of real numbers or as a set of complex numbers. A point cc is in the interior of a set SS of complex numbers if the entire disk Bϵ(c)Bϵ(c) of radius ϵϵ around cc is contained in S.S. While, a point cc belongs to the interior of a set SS of real numbers if the entire interval (c-ϵ,c+ϵ)(c-ϵ,c+ϵ) is contained in S.S. Hence, in the following definition, we will be careful to distinguish between the cases that ff is a function of a real variable or is a function of a complex variable.

Collection Navigation

Content actions

Download module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks