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Use continuity t0 evaluate the limitlim(1,y)->(4,4)x + y6r'...

Question

Use continuity t0 evaluate the limitlim(1,y)->(4,4)x + y6r'

Use continuity t0 evaluate the limit lim (1,y)->(4,4) x + y 6r'



Answers

Use limit laws and continuity properties to evaluate the limit. $$ \lim _{(x, y) \rightarrow(1,3)}\left(4 x y^{2}-x\right) $$

Hello. This is problem in 11 31. Use continuity to evaluate that limit again. If we left f of X to be dysfunction inside a limit, we need to consider what is the domain of disruption and receive that it is an exponents function and so that no man is actually everything in our or you can write negative infinity to infinity, whateveryou wantto refer to write and on that domain sari Women on this domain thief function after vaccine continuous, it's continuous. Therefore, when you eventually the limit exploded one, uh, E to the X Square minus act just replaced that with the would ever facts and because of the continuous so he could just pluck in f one this. The whole reason you could do that is because f is continuous and effort one. So you just talking one e 21 swear minus ones enough to eat. It is zero and that it's just one and that's it. See you in the next radio

In this problem of limit and continue to we have to find the value of limit and also we have to discuss the continuity of the function and we have given that limit ordered pair X. Y and Z approaching towards 134. So this is 134 and the funds and age and the root of X plus Y plus that. So this is X plus Y plus there. Now from here we say that the function of X. Y and Z is equal to and the root of X plus Y plus Z. No, we have to find the value of limits. So we have to replace X. Y. Zed with 134. So this will be F of 134. Now we have to replace X with one Y with three and a third with four. Now putting the value. So this way we will be four plus three. That is seven plus one. That is equals to a root of eight. When we solve it. This is equal to two routes to. Now the value of this limited to route to. Now we have to discuss the continuities. So function of X, Y and Z. That is equals two. And the root of X plus Y plus is it? Now this person would be continuous when the Denham here when the under root term that is X plus Y plus that should be greater or equals to zero. That means the condition is X plus Y plus. That should be greater or equals to zero. So we say that the function is continuous. Four here. X plus Y, plus that is greater than zero, so this is the right answer.

Players are really going to start proper number three here, live it X comma by trains to my Nesler call Mark toe, which is given by ex like you divided by X plus my video vehicles to my effort into True Cube turned my wrestler plus two with physical too. Why not said by far part of my escape?

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.


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