# If #f(9)=9# and #f^'(9)=4#, then what is the value of #lim_(x to9)(sqrt(f(x))-3)/(sqrtx -3)?#

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To find the value of ( \lim_{x \to 9} \frac{\sqrt{f(x)} - 3}{\sqrt{x} - 3} ), we can use L'Hôpital's Rule since both the numerator and denominator approach 0 as ( x ) approaches 9.

First, let's find ( f'(x) ) using the given information:

[ f'(9) = 4 ]

Next, we'll apply L'Hôpital's Rule to the expression:

[ \lim_{x \to 9} \frac{\frac{1}{2\sqrt{f(x)}} \cdot f'(x)}{\frac{1}{2\sqrt{x}}} ]

Now, substitute ( x = 9 ) and ( f(9) = 9 ):

[ = \frac{\frac{1}{2\sqrt{9}} \cdot 4}{\frac{1}{2\sqrt{9}}} ]

[ = \frac{\frac{4}{6}}{\frac{1}{6}} ]

[ = \frac{4}{1} ]

[ = 4 ]

So, the value of ( \lim_{x \to 9} \frac{\sqrt{f(x)} - 3}{\sqrt{x} - 3} ) is 4.

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