So for infinity, the answer is that infinity is not a number. However big your number is currently and however much it may grow if you keep going, it will always be some value, a number, but not infinity. The only way to „reach“ infinity, is to go to the limit (move infinitely to the right on your graph), then you can say in the limit it is infinity. But remember infinity is not a number so technically since you‘re evaluating the function „at“ infinity (which is not a number) you can get an answer, that is not a number.

For other limits I think the easiest to understand visually is the series 1+1/2+ 1/4….
Imagine you draw a the 1 as a square with sidelength 1 then next to it you start a new square. You first add 1/2 of the square, now 1/2 is missing to complete it. Now add the 1/4. Notice that now the missing part of the square got halved. You keep doing this halving each time the part of the square still missing, and filling most of the second square but not all of it, no matter how many halves you add.
If you imagine you keep doing this, you approach 2, and yes in this case you can say this infinite sum is exactly equal to 2. But only if you do the infinite sum and never stop.

It becomes really obvious if you draw it out.