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#