What is the limit as h approaches 0 of ((x+h)^{-5}-x^{-5})/h ?

The limit as h approaches 0 of ((x+h)^{-5}-x^{-5})/h is -5/(x^6)

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limit as h approaches 0 of ((x+h)^{-5}-x^{-5})/h

h \to 0 lim (\frac{( x + h ) ^{- 5} - x ^{- 5}}{h})

- \frac{5}{x ^{6}}

Solution steps

h \to 0 lim (\frac{( x + h ) ^{- 5} - x ^{- 5}}{h})

Simplify

\frac{( x + h ) ^{- 5} - x ^{- 5}}{h} : - \frac{3 x ^{4} + 3 h x ^{3} + h ^{2} x ^{2} + x ( h + x ) ^{3} + ( h + x ) ^{4}}{x ^{5} ( h + x ) ^{5}}

= h \to 0 lim (- \frac{3 x ^{4} + 3 h x ^{3} + h ^{2} x ^{2} + x ( h + x ) ^{3} + ( h + x ) ^{4}}{x ^{5} ( h + x ) ^{5}})

Plug in the value

h = 0

= - \frac{3 x ^{4} + 3 \cdot 0 \cdot x ^{3} + 0 ^{2} x ^{2} + x ( 0 + x ) ^{3} + ( 0 + x ) ^{4}}{x ^{5} ( 0 + x ) ^{5}}

Simplify

- \frac{3 x ^{4} + 3 \cdot 0 \cdot x ^{3} + 0 ^{2} x ^{2} + x ( 0 + x ) ^{3} + ( 0 + x ) ^{4}}{x ^{5} ( 0 + x ) ^{5}} : - \frac{5}{x ^{6}}

= - \frac{5}{x ^{6}}

What is the limit as h approaches 0 of ((x+h)^{-5}-x^{-5})/h ?
The limit as h approaches 0 of ((x+h)^{-5}-x^{-5})/h is -5/(x^6)