But in the second version, the branch point is at $u=-1$, but Mathematica still focuses on $u=0$ so it gets sane results.

Note that to study limits in more than one dimension, it does not suffice to study limits along rays (or lines) only.

I think the key difference is that in the first version, the branch point is $t=0$, and Mathematica focuses on the behavior there - which is inherently weird and strange because it's a branch point (and given that, I'm not sure if using the result leads to computing something correct but strange, or something ill-defined). Mathematica evaluates and simplifies f bt, but otherwise it-correctly-cannot obtain any limit and so just spits out another Limit expression. This substitution does not make a qualitative difference in the original) But if I calculate the limit in Wolfram Alpha, I get the following. (I substituted $x^2 = y$ so that wolfram would finish the calculation for me. For that, I use LHopital and the Fundamental Theorem of Calculus, obtaining the following: lim x 0 1 x 2 + 1 e t t d t 3 x 2 lim x 0 e ( x 2 + 1) x 2 + 1 2 x 6 x lim x 0 e ( x 2 + 1) 3 ( x 2 + 1) e 1 3. Some more observations on Mathematica's behavior: it yields

Wolfram mathematica limit