But, hopefully, it was instructive.

But hopefully that satisfies you.

But hopefully this makes sense.

But hopefully you found that pretty interesting.

But hopefully it's second nature to you.

But anyway, hopefully you found this useful.

But hopefully, that doesn't confuse you too much.

But hopefully you have the intuition of why.

But hopefully that gives you a little intuition.

But anyway, hopefully you found that reasonably useful.

But hopefully this will give you a sense.

But hopefully you get this a little bit.

But you should hopefully understand the problems a

It's, it's entirely fictitious, but, but, hopefully will illustrate the idea.

But hopefully that gives you a good sense of things.

I know this is messy, but hopefully you're understanding it.

But, hopefully, this will give you a little deeper understanding.

But hopefully you'll get a better version of the software app.

But this expedition is not over yet, he points out hopefully.

Hopefully, hopefully, they do it the same way.

But it's hopefully if you think about it this way fairly intuitive.

I don't want to confuse you, but hopefully you should remember this.

I know mine aren't exactly, but hopefully, you get the point, right?

But if you watched the last videos, hopefully that hit the point home.

These are just fancy words, but I think hopefully you have the intuition.

I know that's not a great drawing, but hopefully it serves the purpose.

But hopefully you found that a little bit helpful, and not too confusing.

Hopefully that makes sense and hopefully you understand why it connects.

Hopefully for good.

UE Hopefully not.

But that's a good way to help you, hopefully, remember how to do these.

But hopefully this didn't confuse you too much, because the idea is very simple.

(and this will be a little less conventional but hopefully it will make sense).

Hopefully that makes a little bit sense, but there's another hitch on this problem.

And these words are often used synonymously, but hopefully you understand the distinction now.

But this should hopefully give you an tuition for why this property holds, right?

like this, but hopefully, you'll see that the same type of processes are involved.

But anyway, I just wanted to explain that, and hopefully it hasn't confused you.

Hopefully, it'll make everyone richer, hopefully it'll make the game bigger and hopefully it'll give cricket commentators more time in the business.

At least good work, hopefully, at least it's good work hopefully great work.

But I think at this point, hopefully, you are ready to start doing some problems.

But hopefully this gives you a sense of how all of this stuff is related

But anyway, hopefully at this point you have a pretty good sense of separable equations.

But hopefully over the course of this video, you'll understand that they're pretty straightforward ideas.

But we'll do it and hopefully it will help you understand what deductive reasoning is.