Simple Limits question regarding infinity

[nukeman]

Homework Statement

I am doing some squeeze theorem questions, and I always run into things that is something divided by ∞, or something divided by √∞

Why is it always zero?

like... -3/√∞ = 0 or -1/∞ = 0

why??

Homework Equations

The Attempt at a Solution

 

[STEMucator]

Think about this for a moment. Infinity is not an actual number we can calculate, rather a concept we use to determine what happens over a never ending period of time.

For example consider :

$$lim_{x→∞} \frac{1}{x} = 0$$

So ask yourself for increasing values of x. Say x = 1, 2, 3, ..., n. f(x) is getting smaller and smaller and smaller the bigger and bigger the denominator gets. Notice the values of f(x) → 0 as x → ∞?

So conceptually, some quantity over something really really reeeeally big tends to zero as the bigger thing gets bigger.

Now what about the other case?

$$lim_{x→0^+} \frac{1}{x} = ∞$$

Same sort of argument here. Notice that for positive x = 1/2, 1/3, ..., 1/n, f(x) is getting bigger and bigger the smaller and smaller the denominator gets. So the values of f(x) → ∞ as x → 0.

Does this help?

 

[HallsofIvy]

Homework Statement

I am doing some squeeze theorem questions, and I always run into things that is something divided by ∞, or something divided by √∞

Why is it always zero?

like... -3/√∞ = 0 or -1/∞ = 0

If you "run into things that is something divided by $\infty$", then you are doing the limits incorrectly- I suspect you are trying to "insert" $\infty$ into the formula and you can't do that- "infinity" is not a number.

What you can say is that "limit as x (or n) goes to infinity" is shorthand for "as x (or n) gets larger and larger without bound:
So $\lim_{n\to\infty} \frac{-1}{n}$ means we are looking at what happens as we make the denominator "larger, and larger, and larger, ...}. For example $-1/1000000= -0.000001$, [itex]-1/1000000000= -0.000000001, etc. Now, what do you think would happen if n got even larger and larger?

why??

Homework Equations

The Attempt at a Solution