I'm pretty stuck on this one. I believe the limit is $\displaystyle t^2+1$, but I cannot prove it. The underlying sequence does not appear to be uniformly convergent, which is the only way I'd be able to show the result.Consider the scalar initial value problem

$\displaystyle x'=\frac{t}{1+e^{\mu x^2}}$, $\displaystyle x(0)=1$,

where $\displaystyle \mu\in\mathbb{R}$ is a parameter. Clearly, for each $\displaystyle \mu\in\mathbb{R}$, the solution $\displaystyle x_\mu(t)$ of the initial-value problem exists on $\displaystyle \mathbb{R}$ and is unique. Find

$\displaystyle \displaystyle \lim_{\mu\to 0} x_\mu(t)$

(with justification).

Any help would be much appreciated!