Question

Help me with a limit problem

Answer 1

\[\lim_{r \rightarrow 9}\frac{ \sqrt{r} }{ (r-9)^4 }\]

Answer 2

from the graph it seems to be +infinity

Answer 3

I don't really solve limit problems often so idk how good this advice will be since it doesn't seem to fix the problem just change it into a problem that might be easier or harder.

If you substitute \(r=9 \sin^2 \theta\) then you have this replacing it:

\[\lim_{\theta \to \pi/2} \frac{\sqrt{9\sin^2 \theta}}{(9 \sin^2 \theta - 9)^4} \]

you can use the pythagorean identity and clean it up a bit:
\[= \lim_{\theta \to \pi/2} \frac{1}{3^7} \frac{\sin \theta}{ \cos^8 \theta } \]

I never paid attention in trig when they did all these fancy trig identities so if this can be simplified (probably can) I didn't check but if it can might have your answer.

Good luck.