Are there any limits that have infinity as a value?
Also, as we’ll soon see, these limits may also have infinity as a value. First, let’s note that the set of Facts from the Infinite Limit section also hold if we replace the lim x→c lim x → c with lim x→∞ lim x → ∞ or lim x→−∞ lim x → − ∞ .
When do infinities behave as real numbers do?
Infinities just don’t always behave as real numbers do when it comes to arithmetic. Without more work there is simply no way to know what ∞ − ∞ ∞ − ∞ will be and so we really need to be careful with this kind of problem.
Why do we have an infinity minus an infinity?
We are probably tempted to say that the answer is zero (because we have an infinity minus an infinity) or maybe − ∞ − ∞ (because we’re subtracting two infinities off of one infinity). However, in both cases we’d be wrong. This is one of those indeterminate forms that we first started seeing in a previous section.
What’s the difference between positive infinity and negative infinity?
First, the only difference between these two is that one is going to positive infinity and the other is going to negative infinity. Sometimes this small difference will affect the value of the limit and at other times it won’t.
What do you mean by improper integral from infinity to infinity?
What I guess your professor meant was that lim a → ∞∫a af(x)dx = 0 which is trivially true as the LHS is constantly zero. An improper integral with an endpoint of ∞ means a limit of proper integrals where the endpoint approaches ∞.
What happens when we convert the integral to a limit?
So, the first thing we do is convert the integral to a limit. Now, do the integral and the limit. So, the limit is infinite and so the integral is divergent. If we go back to thinking in terms of area notice that the area under g(x) = 1 x g ( x) = 1 x on the interval [1, ∞) [ 1, ∞) is infinite.