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Problem 36 Easy Difficulty

Use continuity to evaluate the limit.

$ \displaystyle \lim_{x \to \pi} \sin(x + \sin x) $

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we want to find a limit of this function as X approaches pipe, if we can show that this is a continuous function, uh then the limit of this function as X approaches pi is simply going to equal this function evaluated at pie. In other words, if sine of X plus sign effects is a continuous function, then the limit of this function as X approaches pi simply equals the value of this function. When we plug in pi everywhere you see X now off to the side here, I just want to remind us that uh the sine of pi Okay, just low reminders sign of pie is zero. Now, all we need to do before we start plugging in a pie in for X to evaluate this limit, we need to be sure that this is a continuous function. No sign of X is a continuous function. X is a continuous function. So when you add continuous functions you get a continuous function. So X plus sine of X is a continuous function, composite functions. Uh Most of the time our continuous uh sometimes you run into trouble. If you're dividing by a continuous function, you gotta be careful. You're not dividing by zero. Uh There are other times you have to be careful too, but right now uh we're going to be okay. We know that X plus sine of X is a continuous function. Now, sign is a continuous function and we can take the sign of any value on the horizontal axis on the real number line. So regardless of what X plus sine of X comes out to be, we can take the sign of it and uh sign is a continued continuous function. So since sign is continuous and is defined, uh you know, we can take the sign of all values positive, negative. Um We know that X plus sine of X is continuous, we know that sign is continuous. So the sign, this continuous function of this continuous function is going to be continuous. Okay, a continuous function of a continuous function, sine of X plus sign effects will be continuous. Um and we don't have to worry about any places where it's not defined because sign, we can take the sign of any value. So regardless of what X plus sine of X is, we can take the sign of it. So since we determined that this function is continuous. Uh Well, since this is a continuous function, the limit of this function as X approaches pi is simply the value of this function evaluated at high. So the limit of this function as X approaches pi is simply going to be designed of parentheses. Now just substitute pie in for X. We can do that because this was a continuous function X is going to be replaced with pie plus sine of X. X being replaced with pie. Sine of X will be written as a sign of pipe. Now remember sine of pi is zero. So this is really zero. So we have signed parentheses. Hi plus Sine of Pi which we know is zero. Well pi plus zero is pie. So this is really the sine of pi. And we noticed sine of Pi is zero. And so we have arrived at our answer the limit of this function as X approaches, pi is zero.