MHB Does expression equal 0 when value approaches 0?

[tmt1]

If anyone 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?

 

[Greg]

Not necessarily. There is (at least) the indeterminate form $0\times\infty$ to consider.

 

[MarkFL]

Yes, suppose we have:

$$L=\lim_{x\to a}\left(f(x)\cdot g(x)\right)$$

Now, in order to write:

$$L=\lim_{x\to a}\left(f(x)\right)\cdot\lim_{x\to a}\left(g(x)\right)$$

We require that both $$\lim_{x\to a}\left(f(x)\right)$$ and $$\lim_{x\to a}\left(g(x)\right)$$ exist.