Indeterminate forms come from limits. It’s not the question you asked, and I think this answer was a little off the mark because of it. For the sake of shared knowledge, I will explain anyways:
When looking at a limit, it’s important to note that you aren’t working with zero (or infinity, or any number you are studying the limit of), what you are working with are numbers approaching the limit. For example, for (x+1)/(x), the expression has no equivalent value at x=0, as 1/0 does not exist. We can see why if we use the limit as x approaches zero. The numerator will approach 1, and the denominator approaches 0. The numerator has little impact on the value of the expression, but the denominator… dominates the value, for the pun. And, while we can’t evaluate at 0, we can put really small numbers in there and see what happens- and what happens is the expression becomes incredibly large. I’m sure that if you don’t see where this is going, you can go to Desmos or some other graphing calculator and try it for yourself.
As far as the indeterminate form- 0/0 is always undefined, at least in most mathematics. However, if you were to look at equations :
y = x/x
y= x2/x
y= x/x2
you’ll see the curves behaving differently around x=0. The first makes 0/0 look like 1, the second makes 0/0 look like 0, and the last will make 0/0 look like infinity*. Once again, note, however: 0/0 does not exist, and there is discontinuity on all of these curves at x=0.
*Edit: or negative infinity, I forgot that this limit doesn’t exist. Even though the limit doesn’t exist, it is still a useful example.
Indeterminate forms come from limits. It’s not the question you asked, and I think this answer was a little off the mark because of it. For the sake of shared knowledge, I will explain anyways:
When looking at a limit, it’s important to note that you aren’t working with zero (or infinity, or any number you are studying the limit of), what you are working with are numbers approaching the limit. For example, for (x+1)/(x), the expression has no equivalent value at x=0, as 1/0 does not exist. We can see why if we use the limit as x approaches zero. The numerator will approach 1, and the denominator approaches 0. The numerator has little impact on the value of the expression, but the denominator… dominates the value, for the pun. And, while we can’t evaluate at 0, we can put really small numbers in there and see what happens- and what happens is the expression becomes incredibly large. I’m sure that if you don’t see where this is going, you can go to Desmos or some other graphing calculator and try it for yourself.
As far as the indeterminate form- 0/0 is always undefined, at least in most mathematics. However, if you were to look at equations :
you’ll see the curves behaving differently around x=0. The first makes 0/0 look like 1, the second makes 0/0 look like 0, and the last will make 0/0 look like infinity*. Once again, note, however: 0/0 does not exist, and there is discontinuity on all of these curves at x=0.
*Edit: or negative infinity, I forgot that this limit doesn’t exist. Even though the limit doesn’t exist, it is still a useful example.