Question about limit of the summand (modified)

Question

I have the relation $z(t)=\underbrace{y(t)}_{>a>0,\forall t>0}+\underbrace{x(t)}_{>0,\forall t>0}$. It is known that $\lim_{t\to 0}z(t)=a$.

My question is: can we conclude $\lim_{t\to 0}y(t)=a$ and $\lim_{t\to 0}x(t)=0$?

Answer 1

Yes.

If $\lim_{t\rightarrow 0} z(t) = a$, then for every $\epsilon$, there is some $t(\epsilon)$ such that for all $t < t(\epsilon)$, $|z(t) - a| < \epsilon$ (and in particular, $z(t) < a + \epsilon$).

Now, if $z(t) < a + \epsilon$, this both implies that $y(t) < a + \epsilon$ and that $x(t) < \epsilon$ (since $y(t)$ is at least $a$). It follows that since we also have that $y(t) > a$ and $x(t) > 0$, the limits $\lim_{t\rightarrow 0} y(t)$ and $\lim_{t\rightarrow 0} x(t)$ equal $a$ and $0$ respectively.