How do you find the limit of $sqrtr/(r-9)^4$ as r approaches 9?

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Answer 1 · 2017-03-07T13:42:08

$= oo$

For:

$lim_( r to 9) sqrtr/(r-9)^4$

Let $delta = r - 9$ so we are considering the limit:

$lim_( delta to 0) sqrt(delta + 9)/delta^4$

$= lim_( delta to 0) sqrt 9 sqrt(1 + delta/ 9)/delta^4$

A binomial Series on the numerator makes this:

$= lim_( delta to 0) 3 (1 + delta/ 18 + mathcal O(delta^2))/delta^4$

$= lim_( delta to 0) 3/delta^4$

$= oo$

How do you find the limit of sqrtr/(r-9)^4sqrtr/(r-9)^4 as r approaches 9?