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avatar+73 

Let \(f(x) = \sqrt{x} - 5\).

 

Then \(\displaystyle \lim_{ h \rightarrow 0 } \frac{f(7 + h) - f(7)}{h}\) =

 

Find the limit to the math problem.

 Feb 17, 2022
 #1
avatar+23200 
+2

[ f(7 + h) - f(7) ] / h  =  [ (sqrt(7 + h) - 5)  - (sqrt(7) - 5) ] / h

                               =  [ sqrt(7 + h) - 5 - sqrt(7) + 5 ] / h

                               =  [ sqrt(7 + h) - sqrt(7) ] / h

 

Now, multiply both the numerator and denominator by the conjugate of the numerator.

 

           =  [ sqrt(7 + h) - sqrt(7) ] / h   ·   [ sqrt(7 + h) + sqrt(7) ] / [ sqrt(7 + h) + sqrt(7) ]

 

           =  [ 7 + h - 7 ] / [ h · [ sqrt(7 + h) + sqrt(7) ] ]

 

           =  h /  [ h · [ sqrt(7 + h) + sqrt(7) ] ]

 

           =  1 / [ sqrt(7 + h) + sqrt(7) ] ]

 

Now, find the limit of this expression.

 Feb 17, 2022
 #2
avatar+73 
-2

The limit of the expression \(\lim _{h\to 0}\left(\frac{1}{\sqrt{7+h}+\sqrt{7}}\right)\)is \(\frac{1}{2\sqrt{7}}\).

 

The answer to the math problem is 1/(2√7).

 Feb 17, 2022

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