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Handout Wednesday, 8/27
Today we will introduce the idea of a limit, which is essential for understanding the relationship between average velocity and instantaneous velocity.
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Remember that the average velocity is given by\begin{equation*} AV_{[a,b]} = \frac{s(b) - s(a)}{b - a}\text{.} \end{equation*}This does not say how fast you were going at any particular time, but does give your average speed over the interval from \(a\) to \(b\text{.}\)
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What if the interval is very small? Then you might be going faster or slower at certain points on the interval, but it would be reasonable to assume that your average speed is close to your speed at a specific point in time. After all, how much could your speed change in a millisecond?
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Okay, maybe it does change a lot, even in a millisecond. So just zoom in more. Make the interval even smaller.
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As you make the interval smaller and smaller, look at what happens to the average velocity. Does that average velocity get closer to some value? If so, we say that limit value is the instantaneous velocity.
The concept of a limit makes sense for all sorts of functions, not just the average velocity.
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Let \(f(x) = x^2 + 3\text{.}\) What happens as \(x\) approaches 2? We will write this as \(\lim_{x \to 2} f(x)\text{.}\) We want to evaluate this limit.
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Use Desmos to explore this. Plug in values closer and closer to 2 on both sides. Does the value of the function (the “output”) seem to approach a particular number?
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Of course, \(f(2)\) is just \(2^2 + 3 = 7\text{.}\) So we can say that \(\lim_{x \to 2} f(x) = 7\text{.}\) But this doesn’t always work.
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Try \(f(x) = \frac{\sin(x)}{x}\) and evaluate \(\lim_{x \to 0} f(x)\text{.}\) We cannot just take \(f(0)\) since this does not exist. But maybe the limit does?
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An important point: Look at the definition of the limit. We want to find some value that we can get as close as we want to, just by making the input close enough to \(a\) (the input we are approaching).
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Consider the function \(f(x) = \frac{x^2 - 4}{x - 2}\text{.}\) What happens if we try to evaluate this function at \(x = 2\text{?}\) Here we want to simplify this fraction (try factoring the numerator).
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But don’t we just care about average velocity? Well, the function above is an average velocity function. What is the average velocity between \(0\) and \(x\) for the function \(s(x) = x^2\text{?}\)
