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The $ signum $ (or sign) $ function$ , denoted by sgn, is defined by

sgn $ x = \left\{

\begin{array}{ll}

-1 & \mbox{if $ x < 0 $}\\

0 & \mbox{if $ x = 0 $}\\

1 & \mbox{if $ x > 0 $}

\end{array} \right.$

(a) Sketch the graph of this function.

(b) Find each of the following limits or explain why it does not exist.

(i) $ \displaystyle \lim_{x \to 0^+}\text{sgn $ x $} $

(ii) $ \displaystyle \lim_{x \to 0^-}\text{sgn $ x $} $

(iii) $ \displaystyle \lim_{x \to 0}\text{sgn $ x $} $

(iv) $ \displaystyle \lim_{x \to 0}| \text{sgn $ x $} | $

a. SEE GRAPH

b. (i) 1, (ii) -1, (iii) does not exist. (iv) 1

c. $$\begin{array}{l}{\lim _{x \rightarrow 2^{-}} f(x)=0} \\ {\lim _{x \rightarrow 2^{+}} f(x)=1}\end{array}$$

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okay with this signem or sine function. The first thing we want to do is graphic. So the sine of X is equal to one when X is created in zero. So for for positive values of X. Uh The function is defined equal one when X is zero, sign of X equals zero. So let's plot that point. Okay when next zero Case Sine of X is zero. Um But for Uh for positive values of x. design of X is um one and uh sine of X is equal to negative one when X is less than zero. No, one thing we should do just to be a little bit more formal is leave an open circle here and an open circle here, meaning that this point is not on this part of the graph. And this point with this open circle is not on this part of graft. And that's because when X is zero, the sine function is also equal to zero. So here's the 00.0. So for X greater than zero, here's what the graph looks like when X equals zero. That's this point on the graph. And uh for X values that are negative less than zero. Uh Here is what the graph for negative values of X look like. Next we wanna look at some limits. So to limit uh of dysfunction, sign of X. As X approaches zero from the positive side. Well, As our X is approaching zero From the positive side, we have this portion of the graph. This portion of the graph has a constant value of one. So as X. So as X is a moving towards zero from the positive side, that's what this means right here. Extra, approaching zero from the positive side, as X approaches zero from the positive side, The sine function stays constant at one. And so the limit is one. Ah Now do notice that when X is zero. Design function is zero, but that doesn't matter when we're talking about a limit. Okay, to limit as X approaches zero doesn't reach zero, it just approaches zero on the positive side. Uh to function was staying constant at one. That's why limit of the function as X approaches zero on the positive side is one. Well, how about if we approach zero from the how about we approached zero from the negative side. So what is the limit of this function As X approaches zero from the negative site? Well, we're going to figure this out in a similar way as X is approaching zero. Okay, here's X equals zero here. As X is approaching zero from the negative side, The function is constant at -1. And so the limit of the function uh Sine of X As X approaches zero from the negative side is negative. Well, how about the limit of the sine function As X Approaches 0? Not from the positive side, Not from the negative side, but from both sides. All right. Um When the limit as X approaches zero, if it exists, uh then it would have to be the same number as when X approaches zero from the positive side. And the same number as when X approaches zero from the negative side. So at the limit of a function as X approaches zero exists. It has to be the same number regardless of whether it's approaching from the positive side or the negative side. Well, we already clearly know that the limit of this function as X approaches zero from the positive side is different from the limit of the function when extra approaches zero from the negative side. Uh So since approaching zero from different sides gives you different limits, the limit as X approaches zero of the function does not exist, so it's undefined or does not exist for the last part of this question. Uh We want to find the limit as X approaches zero of the absolute value of the sine of X function. Remember that absolute value uh turns everything positive. If it was negative, it turns a positive. For example, the absolute value of negative forward is for The absolute value of a positive number stays positive. For example, the absolute value of five stays 5. Uh And if we have zero then obviously the absolute value of zero is zero. Uh The sine of X. The original sine function when X was greater than zero, the sine of X was defined to be one, so the absolute value of one will still be one. Uh So I want to put a little circle around this point, you'll see why a little bit later. Uh But what I do want to do is I want to graph for positive values of X. uh the absolute value of the sine of X is going to be one sine of X. For positive X values was one. So the absolute value of one is still one. So this is what the graph of the absolute value of sine of X for positive X values looks like. Now. Uh the original sine of X function is defined to be zero when X zero. So the absolute value of zero is still zero. So when X zero are the absolute value of zero is still gonna be zero. And so that's this point over here, if I can grab it. All right, so when X0, the absolute value of the sine of X is going to still be zero. Now for negative values of X, the sine function was to find to be negative one. So for negative values of X, the sine of X was defined to be negative one. But the absolute value of -1 is positive one. And so for negative values of X we are going to have uh the absolute value of the sine of X function equal to one. Again, once again the original sine of X function when X was negative was to find to be negative one. But when we take the absolute value of negative one, we get positive one. And that's why now for negative values of X, the graph of the absolute value of sine of X is a positive one. All right back to this limit right here. Okay. Did not mean to do that. All right. Uh Just for this limit, hopefully just won't affect it. Here we go. We want to take the limit of the absolute value of the sine of X function as X approaches zero. Well, as X approaches zero from the positive side, the absolute value of the sine of X function Is staying constant at one. As X approaches zero from the negative side, the absolute value of the sine of X function once again is constant at one. So, since the limit of the absolute value of sine of X function is the same. Regardless of whether X is approaching zero from the positive side or approaching zero from the negative side. Uh This limit is going to exist and is going to equal one. So, the limit as X approaches zero of the absolute value of sine of X, The answer is one. So the limit does exist. Last I want you to note um that even though the limit of this function as X approaches zero exists and does equal one, uh It doesn't mean that that is going to be the same as the value of that function at zero. Okay, the limit of a function uh when X approaches zero does not have to be the same as the value of the function at zero. The value of this function when X was 010, But the limit as X approaches zero of this function is one.