+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.
+23245 [ 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.
+73 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).
