Let f(x) = √x - 5. Find the limit to the math problem.

Question

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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.

GAMEMASTERX40 Feb 17, 2022

geno3141 · Answer 1 · 2022-02-17T12:26:53

[ 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.

GAMEMASTERX40 · Answer 2 · 2022-02-17T01:20:52

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).