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# Investigate the family of functions$f_n(x) = \tanh (n \sin x)$where $n$ is a positive integer. Describe what happens to the graph of $f_n$ when $n$ becomes large.

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All right, let's answer the question. What is the behavior off? Why equals two hyperbolic tangent? Some people call it Van. I like to call it Tancho, so I'm going to read it that way. The can't off and sign effects. How does this thing behave? Okay, let's go ahead and graphic. If n is equal to one, you can see that it's kind of going to resemble a sine function, as you can see right here. So before we actually observe this, we can take a look at one Tansi. Looks like if you recall the tant function resembles that of a inverse tangent function. Very well. It's just that instead, off negative power to two pi over to it actually goes from negative. Want to positive one? Okay. And if you remember what sign functions looks like that's actually the end side effects know that it's simply going to be one of these shapes. And as an increases or decreases, you can see that the amplitude changes like that. Mhm. All right. So as you can imagine, it's going to be some sort of a mixture of thes to where you can plug in, uh, Ensign X into can church and let's see what's gonna happen. Why equals two engine age off n sign of X. We already did the one place. Let's see what happens if it's equal to two. Uh, it kind of looks a little bit block here, right? How about a much bigger number? Like 17? It's getting even Mawr block here. So what's important to remember about tans? Uh, hyperbolic tangent is the fact that the upper and the lower bound of it it's always going to be between negative 1 to 1. So no matter how large the input is going to be, um, tension is going to be between negative 1 to 1. However, because the input is periodic, you see a periodic, um, shape in tange off and sign effects, and that's basically what's happening here. Let's see what happens if we use a very large number. For example, 10 factorial. You can see that the blocks are going to be almost perpendicular right here, and of course it's not. But I can imagine that these kind of functions can be very useful for things such as engineering's and stuff. So I come from a mathematics or statistics backgrounds, but so I've never really worked with this particularly, but I think it's a really cool function. So this is how can of Ensign of X behaves.

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