Limits

I used to tutor calculus. People really struggle with limits and asymptotes. Asymptotes occur where your function is approaching either positive or negative infinity. This happens when the function is “undefined”. An example of this is if there is some x value that would force the function to divide by zero. You can’t divide by zero. It’s undefined. So you get an asymptote and an infinite limit. Limits help you visualize a function by better understanding how it behaves. You need to check the behavior on both sides of an asymptote - what is the limit when you approach from the left? From the right? They don’t necessarily match and whether or not they do will tell you about the shape of your function. In my picture above, the function in question is barreling down to negative infinity at its asymptote. It looks like the curves on either side of the asymptote will fall down together infinitely until they meet at some “infinity”. Keep in mind that infinity is not a number - it’s more like a direction. Those curves will never touch. They’ll get close - infinitely close - but they will never ever touch.