If any one value in an expression is approaching 0, does the entire expression equal 0?
So for example, for the limit $\lim_{{z}\to{q}} z ln(z) f(z)$. If $\lim_{{z}\to{q}}f(z) = 0$, then does $\lim_{{z}\to{q}} z ln(z) f(z)$ equal 0?
If any one value in an expression is approaching 0, does the entire expression equal 0?
So for example, for the limit $\lim_{{z}\to{q}} z ln(z) f(z)$. If $\lim_{{z}\to{q}}f(z) = 0$, then does $\lim_{{z}\to{q}} z ln(z) f(z)$ equal 0?
Not necessarily. There is (at least) the indeterminate form $0\times\infty$ to consider.
Yes, suppose we have:
$ \displaystyle L=\lim_{x\to a}\left(f(x)\cdot g(x)\right)$
Now, in order to write:
$ \displaystyle L=\lim_{x\to a}\left(f(x)\right)\cdot\lim_{x\to a}\left(g(x)\right)$
We require that both $ \displaystyle \lim_{x\to a}\left(f(x)\right)$ and $ \displaystyle \lim_{x\to a}\left(g(x)\right)$ exist.