An Introduction to Calculus
Calculus is by no means an easy course.
It is the combination and synthesis of a lifetime of learning, and its very name instills fear and dread in high school students worldwide. When I took calculus for the first time, I was genuinely astounded by the amount of formula memorization that goes on within the course. Through memorization, students are pushed into a false sense of complete understanding of a topic, as the nuances found in formulas that solve otherwise easy equations are lost in translation. Thus, the student retains no actual useful knowledge, and fails to both fully comprehend and draw conclusions from questions asked. It is for this reason that I write this guide, in order to nurture the values I hope become a mainstay in all calculus courses.
You do not need to be smart to succeed in Calculus. In fact, the only prerequisite needed for this course is some understanding of Algebra, and an intro to Trigonometry. Everything else will be explained to the best of my ability.
Let’s begin with possibly the simplest topic in calculus: A limit.
You may think of a limit as just what the y-value of a graph at a point seems to be, and at its simplest level, that’s all a limit is.
Take a look at this graph:
Let’s start with a simple example. By asking the question “what is the limit as x approaches 6”, you can interpret that as “what does the y-value of the graph seem to be at x=6?”
And if that seems really easy, it’s because it is. It looks like as the x-value gets really close to 6, the y-value seems to be 4. For future reference, this “gets really close” term is usually said as “approaches”
(Eg. as x approaches 6)
So the “limit as x approaches 6” is just 4. Or, if you want to write it in mathematical form:
Where “x→6” represents what x approaches, “f(x)” represents the function, and where “4” represents your answer. “Limit” is abbreviated to “lim”.
Congratulations, you just found a limit. Try finding the limit as x approaches 8, 2, or 1. (Answers: about 2, 0, and 1 respectively.)
But, you ask, “what about the limit as x approaches 4? How am I supposed to know what the y-value approaches? It looks like the function is defined for all real numbers, but how am I supposed to draw a conclusion if, at 4, the y-value seems to approach both 6 and 3?”
The short answer is that the limit as x approaches 4 doesn’t exist because of what you mentioned. The long answer is also that the limit as x approaches 4 doesn’t exist, but for a different reason. Let’s go with the short answer for now because it’s more intuitive. We’ll discuss the long one later.
Just remember for now that in scenarios like the one above, when things don’t seem to make sense, the limit as x approaches that value does not exist, or simply written as DNE. We’ll justify why in a later article.
This is the beginning of a series of articles attempting to explain the entirety of Calculus AB to a mainstream audience. It is designed to be easily understood, without resorting to dumbing down complex and nuanced topics. A full article list can be found on my profile.
