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

Answer 1

Charles Anctil

#= 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#

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Answer 2

Emily Ammerman

To find the limit of sqrt(r)/(r-9)^4 as r approaches 9, we can use the concept of L'Hôpital's Rule. By applying this rule, we differentiate the numerator and denominator separately and then evaluate the limit again.

Differentiating the numerator, we get 1/2√r. Differentiating the denominator, we get 4(r-9)^3.

Now, we can evaluate the limit again by substituting r=9 into the differentiated numerator and denominator.

Substituting r=9 into the differentiated numerator, we get 1/2√9 = 1/6. Substituting r=9 into the differentiated denominator, we get 4(9-9)^3 = 0.

Therefore, the limit of sqrt(r)/(r-9)^4 as r approaches 9 is 1/6.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.