Why does 0 divided by 0 equal infinity?

Answer 1

Undefined: You cannot divide by zero

Answer 2

0 ÷ 0 is NOT infinity; it is indeterminant.

0 ÷ 0 is any and every value; it is a basis of calculus:

Consider a curve y = some function of x (ie y = f(x), eg y = x²)

Draw a chord on that curve from some value of x to another value of x a little way to its right, ie draw the chord between the two y values at x and x+h where h is a small value.

The slope of this chord is given by change_in_y/change_in_x = (f(x+h)-f(x))/((x+h)-x) = (f(x+h)-f(x))/h

As h gets smaller and smaller, the chord tends towards the tangent of the curve at that x value

and so the slope of the chord tends towards the slope of the tangent at that x value

As h tends towards 0, the value of the slope tends towards some value which is different for different values of x. For y=x² at x=0, the tangent has a slope of 0, but at x=1 the tangent has a slope of 2.

At the limit when h=0, the slope of the chord is the slope of the tangent, but when h=0, the slope is (f(x+0)-f(x))/0 = (f(x)-f(x))/0 = 0/0

Thus 0/0 is both 0 and 2 when considering the slopes of tangents to y=x² at x=0 and x=1.

Answer 3

0 divided by 0 is not infinity: it is not defined.