$SPAD/src/input limit.input

Timothy Daly December 3, 2016

Abstract

Exercise the limit function.

Contents

2

1 Limit of a Function

If f is a function, then

lim_{x−>a} f(x) = A

if the value of f(x) gets arbitrarily close to A as x gets arbitrarily close to a. For example,

lim_{x−>3} x^{2} = 9

since x^{2} gets arbitrarily close to 9 as x approaches 3.

By definition, the limit

lim_{x−>a} f(x) = A

if and only if, for any chosen positive number ϵ, however small, there exists a positive number δ such that

0 < |x − a| < δ implies |f(x) − A| < ϵ

Note that f(x) does not need to be defined at a.

— * —

)set break resume
)spool limit.output
)set message test on
)set message auto off
)clear all
--S 1 of 15
limit((x^2-4)/(x-2),x=2)
--R
--R
--R   (1)  4
--R            Type: Union(OrderedCompletion(Fraction(Polynomial(Integer))),...)
--E 1

———-

2 Right and Left Limits

The limit

lim_{x−>a^{−}} f(x) = A

given that f is defined in the interval (c, a) and f(x) approaches A as x approaches a through the values less than a, that is, as x approaches a from the left.

Similarly

lim_{x−>a^{+}} f(x) = A

means that f is defined in some interval (a, d) and f(x) approaches A as x approaches a from the right.

If f is defined in an interval to the left of a and in an interval to the right of a then the statement

lim_{x−>a} f(x) = A

is equivalent to the conjunction of the two statements

lim_{x−>a^{−}} f(x) = A and lim_{x−>a^{+}} f(x) = A

.

The existence of the limit from the left does not imply the existence of the limit from the right. The existence of the limit from the right does not imply the existence of the limit from the left.

When a function is defined only on one side of a point a, then

lim_{x−>a} f(x)

is identical with the one-sided limit, if it exists. A two-sided limit might not exist because it is not defined outside a certain bound. For example, if

f(x) = \sqrt{x}

then f is defined only at and to the right of 0. Hence,

lim_{x−>0} \sqrt{x} = lim_{x−>0^{+}} \sqrt{x} = 0

But the limit

lim_{x−>0^{−}} \sqrt{x}

does not exist since \sqrt{x} is not defined for real values of x.

A limit might not exist because the function grows without bound as it approaches the limit point. For example, the function \sqrt{\frac{1}{x}} is only defined when x > 0. So

lim_{x−>0^{+}} \sqrt{\frac{1}{x}}

does not exist since \frac{1}{x} gets larger as x− > 0 from the right. So

lim_{x−>0} \sqrt{\frac{1}{x}}

does not exist.

— * —

--S 2 of 15
limit(sqrt(9-x^2),x=-4)
--R
--R

--R   (2)  "failed"
--R                                                    Type: Union("failed",...)
--E 2
--S 3 of 15
limit(sqrt(9-x^2),x=-3)
--R
--R
--R   (3)  [leftHandLimit= "failed",rightHandLimit= 0]
--IType: Union(Record(leftHandLimit: Union(OrderedCompletion(...
--E 3
--S 4 of 15
limit(sqrt(9-x^2),x=-2)
--R
--R
--R         +-+
--R   (4)  \|5
--R                      Type: Union(OrderedCompletion(Expression(Integer)),...)
--E 4
--S 5 of 15
limit(sqrt(9-x^2),x=0)
--R
--R
--R   (5)  3
--R                      Type: Union(OrderedCompletion(Expression(Integer)),...)
--E 5
--S 6 of 15
limit(sqrt(9-x^2),x=2)
--R
--R
--R         +-+
--R   (6)  \|5
--R                      Type: Union(OrderedCompletion(Expression(Integer)),...)
--E 6
--S 7 of 15
limit(sqrt(9-x^2),x=3)
--R
--R
--R   (7)  [leftHandLimit= 0,rightHandLimit= "failed"]
--IType: Union(Record(leftHandLimit: Union(OrderedCompletion(...
--E 7
--S 8 of 15
limit(sqrt(9-x^2),x=4)
--R
--R

———-

3 Theorems on Limits

If

f(x) = c

where c is a constant, then

lim_{x−>a} f(x) = c

Assume that

lim_{x−>a} f(x) = A and lim_{x−>a} g(x) = B

then the limit of a constant times a function is the constant times limit of a function is the constant times the limit.

lim_{x−>a} cf(x) = c ∗ lim_{x−>a} f(x) = cA

The limit of a function plus (or minus) another function is the limit of the first function plus (or minus) the limit of the second function is the sum (or difference) of their limits.

lim_{x−>a} [f(x) ± g(x)] = lim_{x−>a} f(x) ± lim_{x−>a} g(x) = A ± B

The limit of the product of functions is the limit of the first function times the limit of the second function is the product of their limits.

lim_{x−>a} [f(x)g(x)] = lim_{x−>a} f(x) ∗ lim_{x−>a} g(x) = A ∗ B

The limit of the ratio of two functions is the the limit of the first function divided by the limit of the second function provided the limit of the second function is non-zero.

lim_{x−>a} (\frac{f(x)}{g(x)}) = \begin{matrix}lim & x−>a & f(x) \\ lim & x−>a & g(x) \end{matrix} = \frac{A}{B} , if B ≠ 0

The limit of the n^{th} root of a function is the n^{th} root of the limit of the function provided the n^{th} root exists.

lim_{x−>a} \sqrt[n]{f(x)} = ^{n} \sqrt{lim_{x−>a} f(x)} = ^{n} \sqrt{A}, if ^{n} \sqrt{A} is defined

The returned limit seems to be wrong:

4 Infinity

Let

lim_{x−>a} f(x) = +∞

mean that, as x approaches a, f(x) eventually becomes and therefore remains greater than any preassigned positive number, however large. In such a case, we say that f(x) approaches +∞ as x approaches a. The

lim_{x−>a} f(x) = +∞

if and only if, for any positive number M, there exists a positive number δ such that, whenever

0 < |x − a| < δ then f(x) > M

Similarly, let

lim_{x−>a} f(x) = −∞

mean that, as x approaches a, f(x) eventually becomes and thereafter remains less than any preassigned negative number. In that case, we say that f(x) approaches −∞ as x approaches a.

Let

lim_{x−>a} f(x) = ∞

mean that, as x approaches a, |f(x)| eventually becomes and thereafter remains greater than any preassigned positive number. Hence,

lim_{x−>a} f(x) = ∞

if and only if

lim_{x−>a} |f(x)| = +∞

These definitions can be extended to one-sided limits.

— * —

--S 9 of 15
limit(1/x^2,x=0)
--R
--R
--R   (9)   + infinity
--R            Type: Union(OrderedCompletion(Fraction(Polynomial(Integer))),...)
--E 9
--S 10 of 15
limit(-1/(x-1)^2,x=1)
--R
--R
--R   (10)  - infinity
--R            Type: Union(OrderedCompletion(Fraction(Polynomial(Integer))),...)

--E 10
--S 11 of 15
limit(1/x,x=0)
--R
--R
--R   (11)  [leftHandLimit= - infinity,rightHandLimit=  + infinity]
--IType: Union(Record(leftHandLimit: Union(OrderedCompletion(...
--E 11

———-

The limit concepts already introduced can be extended to the case where the variable approaches +∞ or −∞. For example,

lim_{x−>+∞} f(x) = A

means that f(x) approaches A as x− > +∞ or, in more precise terms, given any positive ϵ, there exists a number N such that, whenever x > N, then

|f(x) − A| < ϵ

Similar statements can be made for:

lim_{x−>−∞} f(x) = A

lim_{x−>+∞} f(x) = +∞

lim_{x−>−∞} f(x) = −∞

lim_{x−>+∞} f(x) = −∞

lim_{x−>−∞} f(x) = +∞

— * —

--S 12 of 15
limit(1/x,x=%plusInfinity)
--R
--R
--R   (12)  0
--R            Type: Union(OrderedCompletion(Fraction(Polynomial(Integer))),...)
--E 12
--S 13 of 15
limit(2+(1/x^2),x=%plusInfinity)
--R
--R
--R   (13)  2

--R            Type: Union(OrderedCompletion(Fraction(Polynomial(Integer))),...)
--E 13
)clear all
--S 14 of 15
f := exp(n) * (sin(1/n + exp(-n)) - sin(1/n))
--R
--R
--R                   - n
--R          n    n %e    + 1      n    1
--R   (1)  %e sin(-----------) - %e sin(-)
--R                    n                n
--R                                                    Type: Expression(Integer)
--E 14
--S 15 of 15
limit(f,n=%plusInfinity)
--R
--R
--R   (2)  "failed"
--R                                                    Type: Union("failed",...)
--E 15
)spool
)lisp (bye)

———-

References

[1] Ayres, Frank Jr. and Mendelson, Elliott “Calculus” Schaum’s Outlines 4th edition 1999 ISBN 0-07-041973-6 pp61-63