(x/(x·(x + h)) - (x + h)/(x·(x + h)))/h - Wolfram|Alpha

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Input

(x/(x (x + h)) - (x + h)/(x (x + h)))/h

Result

(1/(h + x) - 1/x)/h

Alternate form

-1/(x (h + x))

Expanded form

1/(h^2 + h x) - 1/(h x)

1/(h (h + x)) - 1/(h x)

-1/(h x + x^2)

Roots

(no roots exist)

Properties as a real function

Domain

${x element R : x!=0 and h x + x^2!=0}$

Range

${y element R : y!=0 and h^2 y^2>=4 y}$

Series expansion at x=0

-1/(h x) + 1/h^2 - x/h^3 + x^2/h^4 - x^3/h^5 + O(x^4)
(Laurent series)

Big‐O notation »

Series expansion at x=∞

-(1/x)^2 + h/x^3 - h^2/x^4 + h^3/x^5 + O((1/x)^6)
(Laurent series)

Big‐O notation »

Derivative

d/dx((x/(x (x + h)) - (x + h)/(x (x + h)))/h) = (h + 2 x)/(x^2 (h + x)^2)

Indefinite integral

integral(-1/x + 1/(h + x))/h dx = (log(h + x) - log(x))/h + constant
(assuming a complex-valued logarithm)

Limit

lim_(x-> ± ∞) (-1/x + 1/(h + x))/h = 0

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