Click here to purchase the entire book in PDF format. Limit Let's pretend that we're standing in a concert hall with a very long reverb time. If I clap my hands and create a sound, it starts dying away as soon as I've stopped making the noise. Each second that goes by, the sound in the concert hall gets closer and closer to no sound. One interesting thing about this model is that, if it were true, the sound level would be reduced each second and, since half of something can never equal nothing, there will be sound in the room forever, always getting quieter and quieter. The level of sound is always getting closer and closer to 0, but it never actually gets there. This is the idea behind a limit. In this particular example, the limit of the sound pressure level in the decaying reverb is 0 - we never get to 0, but we always get closer to it. There are lots of things in nature that follow this idea of a limit. The radioactive half-life of a material is one example. If a radioactive substance loses half of it's radioactivity in one year, then the next year it will lose half of the remaining amount, and the next year, half that amount. So, each year it loses less and less of the original amount. One implication of this is that it never gets to a state of ``no radioactivity'' because half of something is never nothing.2.5 Just remember that a limit is a boundary that is never reached, but you can always get closer to it. Think about Equation 1.69. $$y = \frac{1}{x}$$ (2.69) This is a pretty simple equation that says that $y$ is inversely proportional to $x$. Therefore, if $x$ gets bigger, $y$ gets smaller. For example, if we calculate the value of $y$ in this equation for a number of values of $x$, we'll get a graph that looks like Figure 1.17. Figure 1.17: A graph of Equation 1.69 As $x$ gets bigger and bigger, $y$ will get closer and closer to 0, but it will never reach it. If $x = \infty$ then $y = 0$, but you don't have an $\infty$ button on your calculator. If $x$ is less than $\infty$ but greater than 0, then $y$ has to be a number that is greater than 0 because 1 divided by something can never be nothing. This is exactly the idea of a limit, the first concept to learn when delving into the world of calculus. It's the number that a function (like $\frac{1}{x}$ for example) gets closer and closer to, but never reaches. In the case of the function $\frac{1}{x}$, its limit is 0 as $x$ approaches $\infty$.) For example, take a look at Equation 1.70. $$z = \lim_{x \rightarrow \infty} \frac{1}{x}$$ (2.70) Equation 1.70 says ``$z$ is equal to the number that the function $\frac{1}{x}$ approaches as $x$ gets closer to $\infty$.'' This does not mean that $x$ will ever get to $\infty$, but that it will forever get closer to it. In the example above, $x$ is getting closer and closer to $\infty$ but this isn't always the case in a limit. For example, Equation 1.71 shows that you can have a limit where a number is getting closer and closer to something other than $\infty$. $$\lim_{x \rightarrow 0} \frac{\sin (x)}{x} = 1$$ (2.71) If $x=0$, then we get a nasty number from $f(x)$, but as $x$ approaches 0, then $f(x)$ approaches 1 because $\sin(x)$ gets closer and closer to $x$ as $x$ gets closer and closer to 0. As I said in the introduction, calculus is just math than can cope with infinity. In the case of limits, we're talking about numbers that we get infinitely close to, but never reach.