Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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In Calculus, the term limit is used to describe the value that a function approaches as the input of that function approaches a certain value. There are two ways to demonstrate **Calculus limits**: a numerical approach or a graphical approach. In the numerical approach, we determine the point where the function is undefined and create a table of values to determine the value of the variable as it approaches that point.

I want to talk about limits, limits are really important concept in Calculus, they're in everything in Calculus.

Let's start with the function f of x equals x cubed minus 125 over x-5, then you'll notice that this function is not defined of x=5 but we can still figure out what happens near x=5 and that's what limits are all about. So let's observe I've got I've made a table of values here and I have the inputs for 4, 4.9, 4.99, 4.999. These inputs are approaching five, what are the values doing? Well 61, 73.5 something 74.8 something, you can see that these outputs 9are getting closer and closer to it appears 75.

Now let's see what happens on the other side, so x is coming in towards 5 from the right now, 6, 5.1, 5.01 what's happening to the outputs. 91, 76.51, 75.15 75.015 you can see that here as well the values are getting closer to 75. So you can't plug 5 into this function but you can get as close as you want, and as you get closer and closer to 5 form both sides the value of the function is approaching 75. So here's what we say, we say that f of x approaches 75 as x approaches 5 or another way to write this and this is the way we will commonly express it the limit as x approaches 5 of f of x is 75.