Hello , I have been having trouble with understanding what to do with those ε-δ Limits that involve an extra "x" factor in the result, I'll just write out the problem and perhaps you could correct or instruct me on the logic behind it.

Here, I'm kind of confused. I've seen the best explanation so far here: Created with Camtasia Studio 5 & I'd like to follow the method to get everything in working order.

1:Because of the |x - 2|< δ, (i.e. the x-->2 in the original limit), we know that x is very close to 2, just a bit bigger than it. Because this is so the |x + 3| term is very close to 5, just a bit above it. In fact, it will definitely be less than 6.

We can say with confidence that;

|x + 3||x - 2| < 6|x - 2|

2:Here I'm a bit unsure, can we say that;

6|x - 2| < ε

|x - 2|< ε/6 ? I think this works because although the gap from. say, 5.0000001 to being less than 6 (from the inequality) is big, when we divide ε by 6 we are creating more space.

3:Is there a way to make it look pretty? Is the answer that as long as δ = ε/6 we can guarantee the limit?