tony979 said:
It seems the typical use of epsilon to show limits (that we approach arbitrary closeness) doesn't work here, so I had thought I could say it is the opposite of being bounded, but I'm not sure about how to say this more formally...
The formal definition of an infinite limit states that the limit of a function as the input approaches a certain value is infinite if the output of the function approaches positive or negative infinity as the input gets closer and closer to that value.
An infinite limit is different from a finite limit in that a finite limit has a specific value that the function approaches as the input gets closer to a certain value, whereas an infinite limit has no specific value and instead approaches positive or negative infinity.
Understanding infinite limits is important in calculus and other areas of mathematics as it allows us to analyze the behavior of functions near certain points and make predictions about their behavior. It is also useful in solving various real-world problems involving rates of change and optimization.
To determine if a limit is infinite, you must evaluate the limit as the input approaches the given value from both sides. If the output approaches positive infinity from one side and negative infinity from the other side, then the limit is infinite.
Yes, a function can have more than one infinite limit. This occurs when the function approaches different values of positive or negative infinity as the input approaches different values. For example, the function f(x) = 1/x has infinite limits at x = 0 and x = ∞, as the output approaches positive infinity when x approaches 0 from the right and negative infinity when x approaches ∞.