1. If -1/infinity = infinity or -infinity ?
2. If lim x->infinity^- = infinity & lim x->inifinity^+ = -infinity, then lim x->infinity = does not exist. Am i right? If im wrong please tell me the reason why?
When you say infinity, that is a number which does not exist, and is without bound. It does not exist.
1. Let's start by clarifying the expression -1/infinity. This expression represents the limit of -1 divided by a very large number, approaching infinity. When we divide a negative number by infinity, the result depends on the direction from which we approach infinity.
In this case, -1 divided by a very large positive number would approach 0 from the negative side. Therefore, the answer would be -infinity. On the other hand, if we divide -1 by a very large negative number, it would approach 0 from the positive side, resulting in positive infinity. Therefore, the answer would depend on the direction.
So, to summarize, -1/infinity is undefined because it can have different limits depending on the direction of approaching infinity.
2. You are correct. If the limit as x approaches infinity from the left side (x->infinity^-) is positive infinity, and the limit as x approaches infinity from the right side (x->infinity^+) is negative infinity, then the overall limit as x approaches infinity does not exist.
The reason behind this is that the left-hand limit and the right-hand limit must approach the same value for the overall limit to exist. In this case, the left-hand limit approaches positive infinity and the right-hand limit approaches negative infinity, which means they never converge to the same value. Thus, the overall limit as x approaches infinity is undefined or does not exist.
To determine the existence or non-existence of a limit, you need to check if the left-hand and right-hand limits approach the same value. If they do, then the limit exists. If they approach different values or do not approach any value, then the limit does not exist.