I would like to prove (if so), that the function $f(t,x)=t^{2}e^{nx^2}$ for $t_{0}\in\mathbb{R}$, $n\in\mathbb{Z}$ is Lipschitz with respect to the second variable $x$.
I tried bounding it but I can't seem to do so $$|t^{2}e^{nx^2}-t^{2}e^{ny^2}|=t^2|e^{nx^2}-e^{ny^2}|=2t^2|n\xi e^{n\xi^2}||x-y|$$ Because for large $n$ or $t$ I can't figure out an upper bound.
Anything will help thank you.