def apply(limited_f, limited_g):
from axiom.calculus.is_limited.imply.any_all.limit_definition import of_limited
fx, (x, x0, dir) = of_limited(limited_f, nonzero=True)
gx, (_x, _x0, _dir) = of_limited(limited_g, real=True)
assert dir == _dir
assert x == _x
assert x0 == _x0
return Equal(Limit[x:x0:dir](fx * gx), limited_f.lhs * limited_g.lhs)
def apply(*given):
from axiom.calculus.is_limited.imply.any_all.limit_definition import of_limited
limited_f, limited_g = given
fx, (x, x0, dir) = of_limited(limited_f, real=True)
assert dir == 0
gx, (_x, _x0, dir) = of_limited(limited_g, real=True)
assert dir == 0
assert x == _x
assert x0 == _x0
return Equal(Limit[x:x0](fx + gx), limited_f.lhs + limited_g.lhs)
def apply(given, delta=None, var=None):
from axiom.calculus.is_limited.imply.any_all.limit_definition import of_limited
from axiom.calculus.eq.to.any_all.limit_definition import any_all
fn, (x, x0, dir) = of_limited(given, real=True)
M = fn.generate_var(excludes={x}, var=var, positive=True, real=True)
exists = any_all(Equal(given.lhs, S.Zero), M, delta=delta)
limits = exists.limits + (M, )
return exists.func(exists.function, *limits)
def apply(given, ε=None, δ=None, var=None):
from axiom.calculus.is_limited.imply.any_all.limit_definition import of_limited
from axiom.calculus.eq.to.any_all.limit_definition import any_all
fn, (x, x0, dir), *R = of_limited(given)
# A = given.generate_var(definition=given)
A = fn.generate_var(excludes={x}, **fn.type.dict)
cond = any_all(Equal(given.lhs, A), ε, δ)
B = fn.generate_var(excludes={x}, definition=given.lhs, var=var)
return cond._subs(A, B)
def apply(*given):
from axiom.calculus.is_limited.imply.any_all.limit_definition import of_limited
limited_f, limited_g = given
limited_f = limited_f.of(Equal[0])
fx, (x, x0, dir) = limited_f.of(Limit)
gx, (_x, _x0, _dir), R = of_limited(limited_g)
assert R.is_Interval
assert dir == _dir
assert x == _x
assert x0 == _x0
return Equal(Limit[x:x0:dir](fx * gx), 0)
def apply(is_limited):
from axiom.calculus.is_limited.imply.any_all.limit_definition import of_limited
fx, (x, x0, dir) = of_limited(is_limited, positive=True)
return Contains(Limit[x:x0:dir](log(fx)), Reals)
def apply(is_limited):
from axiom.calculus.is_limited.imply.any_all.limit_definition import of_limited
fx, (x, x0, dir) = of_limited(is_limited, nonzero=True)
return Equal(Limit[x:x0:dir](1 / fx), 1 / is_limited.lhs)