Examples of indeterminate form 0/0

This page is about the indeterminate form 0/0 - a special case of the division by zero.

The files below show functions r(x), g(x) and their quotients m(x)=r(x)/g(x), c(x)=g(x)/r(x).

Derivatives are drawn in the same color as their function, but thinner.

r(0)/g(0) = r'(0)/g'(0)

r(x) = arsinh(x), g(x) = e^x - 1
f(0)/g(0) = 1

r(x) = sin(x), g(x) = sinh(x)

r(x) = tanh(x), g(x) = e^x - 1

r(x) = tanh(x), g(x) = arsinh(x)

r(x) = tanh(x), g(x) = sinh(x)

r(x) = tanh(x), g(x) = x

(coth(x)) / (1/x)
r(x) = coth(x), g(x) = 1/x

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r(x) = e^x - 1, g(x) = (x+1)² - 1

r(x) = e^x - 1, g(x) = cosh(x + arcosh(2)) - 2

r(x) = e^x+1 - e, g(x) = e^x - 1

r(x) = e^x+1 - e, g(x) = 0.5^x+1 - 0.5

r(x) = 2^x+1 - 2, g(x) = 0.5^x+1 - 0.5

r(x) = 2^x+1 - 2, g(x) = 0.5^x - 1

r'(0) = g'(0) = 0     and     r(0)/g(0) = r"(0)/g"(0)

r(x) = cosh(x) - 1, g(x) = sinh²(x)

Functions and quotients

Functions and derivatives

r(x) = cosh(x) - 1, g(x) = sin²(x)

Functions and quotients

Functions and derivatives

r(x) = cos(x)-1, g(x) = x²

Functions and quotients

Functions and derivatives

r(x) = cos(x) - 1, g(x) = sinh²(x)

Functions and quotients

Functions and derivatives

r(x) = cos(x) - 1, g(x) = sin²(x)

Functions and quotients

Functions and derivatives

r(x) = cos(x) - 1, g(x) = cosh(x) - 1

Functions and quotients

Functions and derivatives

r(x) = cos(x) - 1, g(x) = cos²(x) - 1

Functions and quotients

Functions and derivatives