Answer to Question #194333 in Calculus for Moe

Question #194333

True or False.

If f(x)=1/xf(x) = 1/√x , then f(0)f'(0) does not exist.


1
Expert's answer
2021-05-20T11:34:24-0400

f(x)=1xf(x)=\dfrac {1}{\sqrt x} , now to check that derivative exist or not , we know that to check the derivative of function exist or not , we apply formula -


f(x)=lim h0f(x+h)f(x)hf^{'}(x)= _{{lim\ h}{\to}0}\dfrac{f(x+h)-f(x)}{h}

, now limh0_{limh\to0} 1x+h1xh\dfrac {\dfrac {1}{\sqrt {x+h}}-\dfrac{1}{\sqrt x}}{h}


=limh0=_{limh\to0} xx+hxx+hh\dfrac {{ \dfrac {{\sqrt x}-{\sqrt{ x+h}}}{{\sqrt x}{\sqrt {x+h}}}}}{h}


=limh0=_{limh\to0} x+xx+hx+hx(x+h)hxx+h(x+x+h)\dfrac{x+{\sqrt x}{\sqrt {x+h}}-{\sqrt{x+h}{\sqrt {x}}-(x+h)}}{h{\sqrt{x}{\sqrt {x+h}}}({\sqrt{x}+\sqrt{x+h)}}} =1x52=\dfrac {1}{x^{\dfrac {5}{2}}}


hence we can clearly see that f'(0), of a function tends to infinity hence f'(0) does not exist .



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