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Textbook by: Lawrence Baggett. E-mail the author

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:SCf:SC be a function, where SC,SC, 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 zSzS 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 zSzS and |z|B,|z|B, then |f(z)-L|<ϵ.|f(z)-L|<ϵ.

Analogously, if SR,SR, we say limxf(x)=Llimxf(x)=L if for every ϵ>0ϵ>0 there exists a real number BB such that if xSxS and xB,xB, 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 xSxS and xB,xB, 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=limxc-0f(x),L=limxc-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=limxc+0f(x),L=limxc+0f(x), if for every ϵ>0ϵ>0 there exists a δ>0δ>0 such that if xSxS 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:SCf:SC and g:SCg:SC 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 zSzS 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 limzcg(z)0,limzcg(z)0, then
limzcf(z)g(z)=limzcf(z)limzcg(z),limzcf(z)g(z)=limzcf(z)limzcg(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,
limzcf(z)=limzcu(z)+ilimzcv(z).limzcf(z)=limzcu(z)+ilimzcv(z).
(6)

Exercise 1

1. Prove Theorem 1. HINT: Compare with (Reference).
2. Prove that limxcf(x)=Llimxcf(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:SCf:SC is continuous at a point cc of SS if and only if limxcf(x)=f(c).limxcf(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 xS,xS, and that gg is continuous at c.c. Prove that limxcf(x)=g(c).limxcf(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 limx0f(x)limx0f(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 limxcf(x)limxcf(x) exists if and only if the two one-sided limits limxc-0f(x)limxc-0f(x) and limxc+0f(x)limxc+0f(x) exist and are equal.

Exercise 5: Change of variable in a limit

Suppose f:SCf:SC is a function, and that limxcf(x)=L.limxcf(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 limy0g(y)=limxcf(x).limy0g(y)=limxcf(x).
3. Suppose TC,TC, that h:TS,h:TS, and that limydh(y)=c.limydh(y)=c. Prove that
limydf(h(y))=limxcf(x)=L.limydf(h(y))=limxcf(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.

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