Continuity

Karl Weierstrass (1815 – 1897) was the mathematician who (finally) formalized the definition of continuity. In that definition was the definition of limit. So, which came first – continuity or limit? The ideas and situations that required continuity could only be formalized with the concept of limit. So, looking at functions that are and are…

A Family of Functions

Good Question 2, continued.  Today we continue looking at Good Question 2 from our last post. The jumping off place was the BC calculus exam question 5 from 2002. We looked at the differential equation  and its slope field and discussed the questions asked on the exam. Then we saw how to actually solve the equation…

Asymptotes

Horizontal asymptotes are the graphical manifestation of limits as x approaches infinity. Vertical asymptotes are the graphical manifestation of limits equal to infinity (at a finite x-value). Thus, since . The graph will show a horizontal asymptote at y = 1. Since the graph of approaches the x-axis as an asymptote, it follows that . (The fact that this graph…

Why Limits?

There are four important things before calculus and in beginning calculus for which we need the concept of limit. The first is continuity. Most of the time in pre-calculus mathematics and in the calculus we deal with nice continuous functions or functions that are not continuous at just a few points. Limits give us the…