Calculus surrounds all of us every day. But most people either don't want to learn about this glorious subject, or find it too difficult. That is where this book comes in. It will explain what calculus is, and include detailed examples to help you u...
Before we go into the wonderful world of Calculus, we must first understand an easier concept, called a limit.
At first glance, limits may seem kind of daunting, just like Calculus, but it is actually really easy once you get the hang of it. For example, please feast your eyes on the following equation:
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It looks really scary, doesn't it? What a limit means is, what does the expression get closer to, as a variable gets closer to a number?
The above example is simply saying: As the value of x gets closer to infinity, 1÷x gets closer and closer to 0. Of course, ∞ isn't a real number at all. It just means that it gets bigger and bigger. As x gets bigger, the expression gets smaller.
When x=1, 1÷x=1 When x=10, 1÷x=0.1 When x=100, 1÷x=0.01
Here is a table of that:
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It all seems simple now. But there are other limits for us to explore. Like:
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If x is equal to 2, then the expression is equal to 4÷0.
But hey! Why not divide by zero? Wouldn't the answer just be zero? The reason behind this is because division is also known is the inverse of multiplication. So when you are asking, "What does 10÷2 equal?" You are really asking, "2 times what equals 10?"
If we were to apply this to the zero problem, and divide 4 by 0, we are really asking, 0 times what equals 4? Of course the answer can't be zero, because 0 times 0 equals 0. But at the same time the answer can't be 4. The answer is actually ±∞. Before the line reaches an x-value of 2, the y-value is increasing, up until a turning point of (0,0). After that point, the y-value decreases, and gets infinitely lower. You could say that is approaching -∞. Likewise, going backwards from an x-value like 10, the y-value decreases, until a turning point of (4,8). Before that point, the y-value approaches ∞. If this doesn't make sense, please look at the following graph:
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Even if the graph was zoomed out an infinite amount, there is no x-value of 2. There is an x-value of 1.99, to which there is a y-value of -396.01. There is an x-value of 1.99999, to which there is a y-value of -399996.00001. There is even an x-value of 1.99999999999999999999, to which there is a y-value of -399999999999999999996.00000000000000000001. The point is, as you get closer to 2, the expression overall gets closer to either ∞ or -∞, whichever way you are getting closer.
Overall, limits mean that if a variable gets closer to a certain number, what will the expression get closer to. Limits are used a lot in Calculus, especially in Differential Calculus. The main limit used in Differential Calculus is:
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But, we'll get to that next chapter.
For now, here are 4 rules of limits:
1. Limit of Equal Quantities If a and b are always equal to each other, then their limits are equal too.
2. Limit of a Sum A limit of a sum of functions can be rewritten as a sum of limits of individual functions.
3. Limit of a Product A limit of a product of functions can be rewritten as a product of limits of individual functions.
4. Limit of a Quotient A limit of a quotient of functions can be rewritten as a quotient of limits of individual functions.
