How Do You Simplify the Limit of f(x)=√x at x=7?

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SUMMARY

The limit of the function f(x) = √x as x approaches 7 can be simplified using the definition of the derivative. Specifically, the expression lim as h approaches 0 of [f(x+h) - f(x)]/h translates to lim as h approaches 0 of (√(7+h) - √7)/h. To eliminate the square roots in the numerator, rationalization is employed, resulting in the expression (√(7+h) - √7)(√(7+h) + √7)/(h(√(7+h) + √7)). This method effectively simplifies the limit calculation.

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Homework Statement



Use this limit for the following problems: lim as h approaches 0 of [f(x+h)-f(x)]/h.

f(x)=[tex]\sqrt{x}[/tex], x=7


Homework Equations





The Attempt at a Solution



lim as h approaches 0 of [f(x+h)-f(x)]/h

= lim as h approaches 0 of ([tex]\sqrt{7+h}[/tex]-[tex]\sqrt{7}[/tex])/h

I'm having trouble simplifying the numerator, is there any way to get rid of the square roots in the numerator? Thank you.
 
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Rationalize the numerator

[tex]\frac{ (\sqrt{7+h} - \sqrt{7})( \sqrt{7+h} + \sqrt{7})}{h( \sqrt{7+h} + \sqrt{7}}[/tex]
 

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