Question

How do you find `lim (x+1)^(3/2)-x^(3/2)` as `x->oo`?

Answer 1

`oo` or the graph of the function will get undoubtedly large.

Explanation:

So we have `lim_ (x->oo)(x+1)^(3/2)-x^(3/2)`

We can see that this function is continuous when `x>=0`
Remember that when a function is continuous at `c`, then `lim_ (x->c)f(x)=f(c)`
So let's substitute `oo` in the place of `x`.

`(oo+1)^(3/2)-oo^(3/2)`

Now how do we solve that?

Well, let's use logic here.
If there is this really,really large number, and we are raising it to a power greater than one, will get an answer even greater than what we started with. Also, this function will give a positive value for any `x` values that are equal to or greater than one.

Therefore, our function will get undoubtedly large as `x` approaches infinity.

So you can say that `lim_ (x->oo)(x+1)^(3/2)-x^(3/2)` is `oo` or that it gets undoubtedly large.

We can even look at the graph of our function.
graph{(x+1)^(3/2)-x^(3/2) [-10, 10, -5, 5]}
Even though the rate that this is increasing is decreasing, there is no limit of how much the `y` value can be.