Why does $\lim\limits_{x\to-\infty}(\sin x+2)\ln(-x)$ equal $\infty$?
Breaking up the limit:
Since the limit is $DNE \cdot \infty$, why does it equal infinity instead of DNE? Wouldn't the function continue oscillating forever, prohibiting any end limits?
Thanks for the help!
On
We have $\ln(-x)\to +\infty$ when $x\to -\infty$ and $\sin x+2\geq -1+2=1$ hence $(\sin x+2)\ln(-x)\geq \ln (-x)$ which converges to $+\infty$.
On
As with any indeterminate form, you have to look at the actual behavior of the specific function, not just at the behavior of its component parts. As you say, $\ln(-x)$ blows up as $x\to-\infty$. You’re multiplying it by $\sin x+2$, which oscillates over the range $[1,3]$. If $f(x)=(\sin x+2)\ln(-x)$, at each $x<0$ you have $$\ln(-x)\le f(x)\le 3\ln(-x)\;.$$ As $x\to-\infty$, both $\ln(-x)$ and $3\ln(-x)$ increase without bound, and $f(x)$ is trapped between them, so it must also increase without bound: $\lim\limits_{x\to-\infty}f(x)=\infty$.
All you actually need here is the fact that $\ln(-x)$ is blowing up, since $f(x)$ is trapped above it: as $\ln(-x)$ increases without bound, $f(x)$ is forced up as well.
It would be a very different story if your function were $(x\sin x)\ln(-x)$, for instance. The oscillations in $x\sin x$ get bigger and bigger in both directions as $x\to-\infty$, so the oscillations in $(x\sin x)\ln(-x)$ do so as well: $\lim\limits_{x\to-\infty}|(x\sin x)\ln(-x)|=\infty$, but $\lim\limits_{x\to-\infty}(x\sin x)\ln(-x)$ doesn’t exist.
Using $\ln(-x) \geq 0$ for $x \leq -1$, we get (for $x \leq -1$) $$-1 \leq \sin x \leq 1 \\ \Downarrow \\ 1 \leq \sin x + 2 \leq 3 \\ \Downarrow \\ \underbrace{\ln(-x)}_{\stackrel{x \to -\infty}{\longrightarrow} \infty} \leq (\sin x + 2) \ln(-x) \leq \underbrace{3 \ln(-x)}_{\stackrel{x \to -\infty}{\longrightarrow} \infty}$$ So your function is squeezed between two functions that both diverge to $\infty$, and therefore the function itself must also diverge to $\infty$.