gx x evaluate and simplify ga h ga h when a 1 a 2 a 3

g(x) = x, evaluate and simplify

g(a + h) g(a)

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h

when a = 1, a = 2, a = 3, a = x

Solution

Solution: Given g(x) = x,

Therefore \\frac{g(a+h)-g(a)}{h}=\\frac{\\sqrt{a+h}-\\sqrt{a}}{h}

For a = 1,

\\frac{g(1+h)-g(1)}{h}=\\frac{\\sqrt{1+h}-\\sqrt{1}}{h}
=\\frac{\\sqrt{1+h}-1}{h}

=\\frac{({1+h})^{1/2}-1}{h}

=\\frac{({1+\\frac{1}{2}h+\\frac{1}{2}(\\frac{1}{2}-1})\\frac{h^2}{2!}+...-1}{h}

=\\frac{({\\frac{1}{2}h+\\frac{1}{2}(\\frac{1}{2}-1})\\frac{h^2}{2!}+...}{h}

={\\frac{1}{2}+\\frac{1}{2}(\\frac{1}{2}-1})\\frac{h}{2!}+...

Now

For a = 2,

\\frac{g(2+h)-g(2)}{h}=\\frac{\\sqrt{2+h}-\\sqrt{2}}{h}   

=\\frac{({2+h})-{2}}{h(({2+h})^{1/2}+\\sqrt{2})}

=\\frac{1}{((\\sqrt{2+h})+\\sqrt{2})}

For a = 3,

\\frac{g(3+h)-g(3)}{h}=\\frac{\\sqrt{3+h}-\\sqrt{3}}{h}   

=\\frac{({3+h})^{1/2}-{\\sqrt{3}}}{h}.

And

For a = 3,

\\frac{g(x+h)-g(x)}{h}=\\frac{\\sqrt{x+h}-\\sqrt{x}}{h}   

. =\\frac{{x+h}-{x}}{h(\\sqrt{x+h}+\\sqrt{x})}

=\\frac{{h}}{h(\\sqrt{x+h}+\\sqrt{x})}

=\\frac{{1}}{(\\sqrt{x+h}+\\sqrt{x})}