1/0 is not defined and 0/0 is indeterminate

1/0 is not defined and not infinity.

x divided by y means to make "y equal parts of x". Since it is not possible to make 0 parts of 1, 1/0 is not defined.

Proof goes like this
consider the series 1/(1/2) , 1/(1/3) , 1/(1/4) , ...................................
The denominator in the series becomes smaller and smaller and eventually will become zero. Thus eventually we will get a number1/0 in the series.
The series becomes 2 , 3 , 4 , ................................ Hence we can conclude 1/0 = infinity

consider the series 1/(-1/2) , 1/(-1/3) , 1/(-1/4) , ...................................
The denominator in the series will become zero. Thus eventually we will get a number1/0 in the series.
The series becomes -2 , -3 , -4 , ................................ Hence we can conclude 1/0 = -infinity

Hence we get 2 values of 1/0. We are not sure whether 1/o is infinity or -infinity.
Therefore 1/0 is not defined. Anything which doesn't make sense is not defined.

Proof 2
Let 1/0 = x .
Hence 0*x = 1. No value of x satisfies the equation. Hence x is not defined.
Therefore 1/x is not defined.

On the other hand consider 0/0
Let 0/0 = x.
Hence 0*x= 0. Any value of x satisfies the equation. Hence x cannot be determined.
Therefore 0/0 is indeterminate.

Note Indeterminate and not defined are not one and the same. Indeterminate means the solution can take any value. However not defined means the solution doesn't have any value.