## a unique way of thinking

Think of the graph of the function y=f(x) as a roller coaster track. You, as an unlucky amusement park visitor, is thrown into an eternally powered roller coaster that rides on the graph.

The limit of the function at x=a, is what your position is when the shadow of your cart goes over x=a. Of course, if the track is solid at that point, your cart is on the track. However, if there is a small hole on the track--- that is, the function is not defined at x=a, but defined at both sides of it, your cart will fly right over the gap-- hold your breath.

So if the function has the form y=2x, yet x is not defined at 2, then the limit of the function at x=2 is still four.

(By the way, for those who play MW3, Mind the Gap is my favorite campaign mission.)

Now, if the limit goes to infinity-- then you visit the kingdom on the clouds. We say the limit diverges, or it is infinity. Just like the U.S. debt ceiling, if you know what I'm talking about. If there are wild variations-- the cart will get thrown off track-- the limit does not exist! Using the same reasoning, we can see if the function approaches different values from both sides, your cart, alongside with you in it, will undergo a peculiar experience, not available at any licensed amusement park. Chances are you are not going to wake up to tell the tale. The limit does not exist.

The limit of the function at x=a, is what your position is when the shadow of your cart goes over x=a. Of course, if the track is solid at that point, your cart is on the track. However, if there is a small hole on the track--- that is, the function is not defined at x=a, but defined at both sides of it, your cart will fly right over the gap-- hold your breath.

So if the function has the form y=2x, yet x is not defined at 2, then the limit of the function at x=2 is still four.

(By the way, for those who play MW3, Mind the Gap is my favorite campaign mission.)

Now, if the limit goes to infinity-- then you visit the kingdom on the clouds. We say the limit diverges, or it is infinity. Just like the U.S. debt ceiling, if you know what I'm talking about. If there are wild variations-- the cart will get thrown off track-- the limit does not exist! Using the same reasoning, we can see if the function approaches different values from both sides, your cart, alongside with you in it, will undergo a peculiar experience, not available at any licensed amusement park. Chances are you are not going to wake up to tell the tale. The limit does not exist.