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

1 Answer
Feb 19, 2018

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)