Suppose we define a sequence, $a_k$, by $$ \begin{cases} x^2 e^{-\frac{k}{2}} \ \text{if $x \geq 0$} \\ 0 \ \text{otherwise} \end{cases} $$ Then, say that we want to take its limit as $k \to \infty$. My question mainly relates to how we work with $x$. It seems that the limit would depend on the value of $x$, or at the very least we would have two cases to consider. If I'm not mistaken, both are zero, so it seems reasonable to conclude that the limit is $0$. Is this the correct line of thinking? Is there a better way to formalize such an argument?
Thanks.
EDIT: Per a very helpful comment from Bungo, let's consider one alternate case where $x$ is not solely a constant. Say, for example, we define a sequence by $$ \begin{cases} x^2 e^{-\frac{kx}{2}} \ \text{if $x \geq 0$} \\ 0 \ \text{otherwise} \end{cases} $$ In this alternate case, would we proceed differently?
Well, the limit will again be a function of $x$, so in general your answer will involve $x$. For example, the limit as $k$ goes to infinity of $f_k(x)=x^2+x/k$ is $f(x)=x^2$. In your care, the limit is the constant function $0$. In general, one is often concerned with additional properties of the sequence of functions and not just the "pointwise" limit. For example, is the convergence uniform in $x$? (If not, then $f$ may fail to be continuous even if all the $f_k$ are continuous.) Or is the convergent sequence dominated by an $L^1$ function so that you can apply the dominated convergence theorem to its integral?