Limits are needed to define continuity of a function, a function’s derivative, the definite integral (this can actually be done better without limits via Darboux integration, but this is seldom used in calc courses), and sequences/series/power series-i.e., the main elements of the rest of Calc I & II. When you look at the graph of what has just happened, you can see that the speed limit is basically a speed value that your car approaches. Then you accelerate and start driving a bit faster, until you get to the speed limit, at which point you don’t go any faster (because we obey the laws!), and then you approach a stop sign and slow down to come to a stop.
So, imagine then what a graph would look like of the speed of your car over time. Your car is not allowed to travel any faster than this. Let’s say there is a posted speed limit of 50 mph. The concept of a limit is essentially the same as when we’re talking about a speed limit. But how does this construct fit into calculus, or does “limit” mean something else entirely in this context? We all know about limits in terms of, for example, the speed limit for a car or other defined measures.
It’s a fairly intuitive concept, and we will eventually see that it is a very important concept as well! On the other hand, at first glance, the concept of a “ limit” sounds a bit more abstract. It sounds like a math concept that explains how fast something is changing.
To a math student, “rate of change” kind of makes some sense. Back when I was taking my introductory calculus courses, one of the first topics that we covered was a section on limits and rates of change.
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