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SYMBMATH.H06
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7.4 Limits
SymbMath finds real or complex limits, and discontinuity
when x approaches to x=x0 by functions
subs(f, x=x0)
lim(f, x=x0)
First use subs() to find limits, if the result is undefined
(indeterminate forms, e.g. 0/0, inf/inf, 0*inf, and 0^0), then use
lim() to try again; if the result is discont, then use the one-side
limit by c+zero or c-zero.
Example 7.4.1. Find limits of types 0/0 and inf/inf.
Input:
p=(x^2-4)/(2*x-4)
subs(p, x=2)
lim(p, x=2)
subs(p, x=inf)
lim(p, x=inf)
end
Output:
p = (x^2 - 4)/(2 x - 4)
undefined
2
undefined
inf
The "discont" (discontinuity) means that the expression has
a discontinuity and only has the one-sided limit value at x=x0. Users
should use x0+zero or x0-zero to find the one-sided limit. The
f(x0+zero) or f(x0-zero) is the right-sided limit or left-sided limit
as approaching x0 from positive (+inf) or negative (-inf) direction,
respectively, i.e. limit as zero -> 0.
SymbMath find a left-sided or right-sided limit when x
approaches to x0 from positive (+inf) or negative (-inf) direction at
discontinuity by functions
subs(f, x=x0+zero)
subs(f, x=x0-zero)
lim(f, x=x0+zero)
lim(f, x=x0-zero)
Example 7.4.2. Find the left-sided and right-sided limits of
y=exp(1/x), (i.e. when x approaches 0 from positive and negative
directions).
Input:
y=exp(1/x)
subs(y, x=0)
subs(y, x=0+zero)
subs(y, x=0-zero)
end
Output:
y = exp(1/x)
discont
inf
0
The built-in constants of inf or -inf, zero or -zero, and
discont or undefined can be used as numbers in calculation of
expressions or functions.
Example 7.4.3.
Input:
1/sgn(0)
1/sgn(zero)
end
Output:
discont
1