Question

Dada a função f (x) = 1/x, mostrar que f (1 + h) - f (1) = - h/ (1 + h). Calcular f (a + h) - f (a).

Answer 1

$f(x)=\dfrac{1}{x}~~~\therefore~~~\boxed{\boxed{f(1+h)=\dfrac{1}{1+h}}}\\\\\\f(x)=\dfrac{1}{x}~~~\therefore~~~\boxed{\boxed{f(1)=1}}$

Então:

$f(1+h)-f(1)=\dfrac{1}{1+h}-1\\\\\\f(1+h)-f(1)=\dfrac{1}{1+h}-\dfrac{1+h}{1+h}\\\\\\f(1+h)-f(1)=\dfrac{1-(1+h)}{1+h}\\\\\\f(1+h)-f(1)=\dfrac{1-1-h}{1+h}\\\\\\\boxed{\boxed{f(1+h)-f(1)=-\dfrac{h}{1+h}}}$
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$f(a+h)-f(a)=\dfrac{1}{a+h}-\dfrac{1}{a}\\\\\\f(a+h)-f(a)=\dfrac{1\cdot a}{(a+h)\cdot a}-\dfrac{1\cdot(a+h)}{a\cdot(a+h)}\\\\\\f(a+h)-f(a)=\dfrac{a}{a(a+h)}-\dfrac{a+h}{a(a+h)}\\\\\\f(a+h)-f(a)=\dfrac{a-(a+h)}{a(a+h)}\\\\\\f(a+h)-f(a)=\dfrac{a-a-h}{a(a+h)}\\\\\\\boxed{\boxed{f(a+h)-f(a)=-\dfrac{h}{a(a+h)}}}$