$SPAD/src/input triglim.input

Clifton J. Williamson and Timothy Daly

December 3, 2016

Abstract

Contents

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1 License

— license —

--Copyright The Numerical Algorithms Group Limited 1991.

———-

— * —

)set break resume
)spool triglim.output
)set message test on
)set message auto off
)clear all

———-

2 Limits involving trig and inverse trig functions

Some of these limits were taken from bug reports submitted before changes were made to limit and related series packages.

In each case, the argument to ’atan’ has a pole and the left-hand and right-hand limits differ

— * —

--S 1 of 6
limit(atan(1/sin(x)),x = 0)
--R
--R
--R                          %pi                 %pi
--R   (1)  [leftHandLimit= - ---,rightHandLimit= ---]
--R                           2                   2
--IType: Union(Record(leftHandLimit: Union(...
--E 1
--S 2 of 6
limit(atan(sqrt(1 - x**2)/x),x = 0)
--R
--R
--R                          %pi                 %pi
--R   (2)  [leftHandLimit= - ---,rightHandLimit= ---]
--R                           2                   2
--IType: Union(Record(leftHandLimit: Union(...
--E 2
--S 3 of 6
limit(atan(-sin(x)/(cos(x) + a)),x = acos(-a))

--R
--R
--R                          %pi                 %pi
--R   (3)  [leftHandLimit= - ---,rightHandLimit= ---]
--R                           2                   2
--IType: Union(Record(leftHandLimit: Union(...
--E 3
--S 4 of 6
limit(atan(sin(x)/(cos(x) + a)),x = acos(-a))
--R
--R
--R                        %pi                   %pi
--R   (4)  [leftHandLimit= ---,rightHandLimit= - ---]
--R                         2                     2
--IType: Union(Record(leftHandLimit: Union(...
--E 4

———-

We’ll get these next two if sqrt(1 - a**2) is considered to be positive by SIGNEF — * —

--S 5 of 6
limit(atan(1/(cos(x) + a)),x = acos(-a))
--R
--R
--R                        %pi                   %pi
--R   (5)  [leftHandLimit= ---,rightHandLimit= - ---]
--R                         2                     2
--IType: Union(Record(leftHandLimit: Union(...
--E 5
--S 6 of 6
limit(atan(1/(sin(x) + a)),x = asin(-a))
--R
--R
--R                          %pi                 %pi
--R   (6)  [leftHandLimit= - ---,rightHandLimit= ---]
--R                           2                   2
--IType: Union(Record(leftHandLimit: Union(...
--E 6
)spool
)lisp (bye)

———-

References^{[1] nothing}

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