Wrong limit result?

Question

I'm trying to evaluate the following limit:

Limit[(d + 1/2)*Log[d + 1/2] - (d - 1/2)*Log[d - 1/2],d->a] 

Normally, when a tends towards 0.5, the limit is supposed to give 0 (which it does when I put a=0). However, when a=0.49999999999999833` (precisely), it gives me the result "Indeterminate". Why is it giving me that result? Can I do something to avoid it? (It creates problems afterwards)

Answer 1

I'm not sure how Limit works for inexact numbers, but the issue is probably related to:

Limit[Log[d - 1/2], d->0.49999999999999833`]

-∞

There was a change in how machine numbers are handled starting in M11.3, and this change means the output of your limit will be different when comparing M12 and M11.1 and earlier.

Now, Limit works best with exact input. If you want to use inexact input, though, it is probably better to use an extended precision number. So:

Limit[Log[d - 1/2], d->0.49999999999999833`16]

-34.0 + 3.1 I

and:

Limit[(d+1/2)*Log[d+1/2]-(d-1/2)*Log[d-1/2],d->0.49999999999999833`16]

-6.*10^-14 + 5.*10^-15 I

both produce answers more in tune with your expectations.

Answer 2

You don't need to take a limit here, you can just use a multiple dispatch function to "plug the holes" at $a=\pm\frac12$:

f[1/2 | 0.5] = 0;
f[-1/2 | -0.5] = I*π;
f[a_] = (a + 1/2)*Log[a + 1/2] - (a - 1/2)*Log[a - 1/2];

This is also much faster and easier:

f[0.5]
(*    0    *)
f[0.49999999999999833`]
(*    -5.83346*10^-14 + 5.2318*10^-15 I    *)