Question

What is it limit as `xrarr9`? `sqrtr/((r-9)^4)`

Answer 1

the limit does not exist (it diverges)

Explanation:

We seek the limit:

`L = lim_(r rarr 9) sqrt(r)/(r-9)^4`

We note that if we substitute `r=9` then the numerator is `3` and the denominator is `0` so we cannot directly evaluate the limit nor can be apply L'Hôpital's rule.

However, this alone tells us that there is an asymptote of the function `f(r) =sqrt(r)/(r-9)^4` at `x=9` and as such the function becomes infinite (from either side) as we approach `x=9`. Thus we can write (where `~` means behaves asymptotically as).

`lim_(r rarr 9) sqrt(r)/(r-9)^4 \ ~ \ 3/0` as `r rarr 9`

In other words

`lim_(r rarr 9) sqrt(r)/(r-9)^4 rarr oo`

And we conclude that the limit does not exist (it diverges)