Continuous limits involving division by x

[peteroupc]

Take the function (sinh(x)+cosh(x)-1)/4. This function when divided by x, has a limit of 1/4 at 0:

In [77]: cc=(sinh(x)+cosh(x)-1)/4                                                                                                                  

In [78]: limit(cc/x,x,0)                                                                                                                           
Out[78]: 1/4

However, when the limit is taken as "x approaches a symbol" such as z, the result of limit doesn't take into account the fact that z could be 0:

In [79]: limit(cc/x,x,z)                                                                                                                           
Out[79]: 
 z    
ℯ  - 1
──────
 4⋅z  

In [80]: limit(cc/x,x,z).subs(z,0)                                                                                                                 
Out[80]: nan

In this case the correct solution to the limit would be a piecewise function:

• 1/4 if z = 0.
• (exp(z) - 1)/(4*z) otherwise.

[anutosh491]

well sinh(x) + cosh(x) is nothing but exp(x) and something like limit((exp(x) -1)/x, x, 0) = 1 is one of the most standard limits you'll find , so i'll try to address this as soon as I can .

>>> limit((exp(x)-1)/x, x, z) # Not the correct output 
(exp(z) - 1)/z
>>> limit((exp(x)-1)/x, x, z).subs(z, 0)
nan

f(x) = (exp(x) - 1)/x is not defined at 0 and hence returns nan here
Well this seems to be a general case rather than just something related to exp(x)

>>> limit(ln(1+x)/x, x, z)
log(z + 1)/z
>>> limit(ln(1+x)/x, x, z).subs(z, 0) 
nan
>>> limit(sin(x)/x, x, z)
sin(z)/z
>>> limit(sin(x)/x, x, z).subs(z, 0)
nan

EDIT: So throughout the gruntz.py , the workflow for these cases doesn't suggest any kind check for singularities! will have to go through that

EDIT 2: Heyy @oscarbenjamin , @jksuom would something like this work(in the sense that I'm not sure which cases would I be missing if any and what else should I be addressing)

diff --git a/sympy/series/gruntz.py b/sympy/series/gruntz.py
index 1154d40b17..fc35b0d9db 100644
--- a/sympy/series/gruntz.py
+++ b/sympy/series/gruntz.py
@@ -701,6 +701,14 @@ def gruntz(e, z, z0, dir="+"):
     except ValueError:
         r = limitinf(e0, z, leadsimp=True)

+    from sympy import singularities, Eq, Piecewise
+    if z0.is_symbol and not str(singularities(r, z0)) == 'EmptySet':
+        piecewise_list = []
+        for singularity in singularities(r, z0):
+            piecewise_list.append((gruntz(r, z0, singularity),Eq(z0, singularity)))
+        piecewise_list += [(r,True)]
+        r = Piecewise(*piecewise_list)
+
     # This is a bit of a heuristic for nice results... we always rewrite
     # tractable functions in terms of familiar intractable ones.
     # It might be nicer to rewrite the exactly to what they were initially,

This works correctly as per what I could see , as I said not at all sure what I could be missing !!! I know that some cases like limit(ln(1+x)/x ,x, z) won't be using gruntz.py and would use only limits.py so a similar fix there at the appropriate place might do it but other than that if you'll have something I would be glad to know .

>>> limit((sinh(x) + cosh(x) -1)/(4*x), x, z)
Piecewise((1/4, Eq(z, 0)), ((exp(z) - 1)/(4*z), True))

>>> limit(sin(x)/x, x, z)
Piecewise((1, Eq(z, 0)), (sin(z)/z, True))

>>> limit((1- cos(x))/x**2, x, z)
Piecewise((1/2, Eq(z, 0)), (-(cos(z) - 1)/z**2, True))