use-the-definition-of-continuity-and-the-properties-of-limits-to-show-that-the-function-is-contin-17

Question

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number.
$$f(x)=\frac{x \sqrt{x}}{(x-6)^{2}} \quad 	ext { at } x=36$$

Julian Gerber · Accepted Answer

So we have the function f of X, and we want to determine whether F of X is continuous at X equals 36. So, first of all, let&#x27;s find out what f of 36 is. Make sure it&#x27;s defined. So f of 36 equals 36 times. Route 36 over 36 minus six squared. We simplify that. Route 36 is just six. So you have 30 to 6 times six, which is, um, 2 16 over 36 months. Six is 30 30 squared is 900. So we have to 16 over 900 which can be simplified, but it doesn&#x27;t matter. Next, we have to find out that the limit exists. So the limit as X approaches 36 of F of X. And we don&#x27;t have to worry about left hand and right hand limits because as the um, the function is the same. It&#x27;s the same function, um, both to the left of 36 to the right of 36. So it&#x27;s gonna be approaching the same limit value. And when we substitute in 36 for acts are answer isn&#x27;t indeterminant. It&#x27;s not 0/0 or infinity over infinity. It&#x27;s a constant values are. Limit is just that same constant value. So we&#x27;ve determined that the limit exists and that it&#x27;s equal to the function. Value F of 36 therefore F of X is continuous at X equals 36.

Julian Gerber · Answer

So we have the function f of X and we want to determine whether or not F of X is continuous at X equals two. So to do that will first find what f of two is. Make sure it exists. So if we just sub student for two, we get two squared minus nine mops. Not sure what happened there. Two squared minus nine over two squared, minus two minus six times two squared. Plus six times too plus nine. So we have four minus nine. And the numerator, which is gonna be negative. Five over two squared is four, um, minus two minus six times four plus six times two is 12 plus nine equal to negative five over four months to his negative too. Minus six is negative. Eight times. Ah, 16 plus nine is 25. So that is going to be equal. Teoh 1/40. Um, right, Yeah, 1/40. So we know effort to exist, first of all, so that&#x27;s good. Our first criteria is filled. Now we need to find the limit as ah f of X approaches. X equals two. So limit comes X approaches two of f of X now because as X approaches to, um f of X will approach f of to weaken, do the same thing. We did hear of substituting, um, in two for X. So if we said student to we&#x27;re just gonna get the same value 1/40 and we don&#x27;t have to worry about left hand and right hand ah, limits because it&#x27;s the same function on both sides. There&#x27;s no, uh, jump. So, um, we see that 1/40 obviously equals 1/40 so it is f of X is continuous at X equals to. So remember, if you have a limit, if you could just plug in the value that you ah are approaching into f of X or your function, Um and you don&#x27;t get something indeterminant like 0/0 or infinity over infinity. If you get an actual value than that&#x27;s just your limit value and because it equals the function value 1/40 we know it&#x27;s continuous

Agata Pavone · Answer

let&#x27;s know prove that the function after backs the function and form X so remind it&#x27;s a no for life. The function off of eggs, equal squares Ruto six then minus X square is continuous this time on the interment minus four for so now we have to prove continue t on and in Terrible And what does it mean on an interval? It miss that the function has to be continuous. Okay, at each point inside is intermittent and this is for sure, true, because the function after Max Square root of this quantity is defined. So let&#x27;s see. First of all, the function and dramatics he&#x27;s defined for it. It&#x27;s ex belonging to this in terms. Why? Because this is just the me minus 44 is just the do mean off this function, and you can easily see why, because the domain of this square root can be easily calculated by survey these inequalities 16 minus x squared Greater oh, equal seal on distributions on just or the values of X in between minus four and four, including for minors for four included. So the function is defined at each point inside this interval on and the name it. Have this function after bags. Um, atop square root off 16 minus x squared. When X approaches anybody you that&#x27;s court, the value A and that say that a belongs a belongs to this in terminal minus for four. So let&#x27;s see the limit while expert to say off the square root equal. So the limit off square root is equal to the square root after limits. So limit off sixth in minus X square eggs a purchase. Hey, this is this is a property of limits so equal. Well, now we have the limit When X approaches. Okay. 0/6 in minus X squared. But this is equal to limit its the limit of a son and on fabric some. So this is equal to the limit. When x purchase a 0/6 in minus. Name it one x they purchase. Okay. Oh, ex queer. So sequin square. Root off. This is the remake of the cost Entous. Just the constant. The same one. So six, then and then who minus limit my neck? Support to say is your X squared is just a square, so you can easily see that this is equal to the value off F at the point A for any body of a belonging to the interval, minus four for closed in terms. Soon the function is continues on thinking about the end points when it as we want, the function continues on the N terminal. Well, we can say that the function will be continuous at X equal minus four will be continues, but only from the right. Only from the right and we&#x27;d be continuous at X equal four, but only from left. Why? Because of course, the function is not defined when X is less than minus four or X is greater than four. So we have continued it&#x27;d but only from the right at the left endpoint and only from the left at right and point.

Daniel Jaimes · Answer

This is from number fourteen. His tour Calculus, a sedition section two point find. Use the definition of continuity and the properties of Clement&#x27;s to show that the function is continuous at the Cuban number, eh? If X is equal to three x to the fourth power minus five x plus the cube root of the quantity X x squared plus four for a close to two on. So the general definition for continuity is that element as X approaches A for the function. Yeah, is equal to F A. And so the way that will show this is, too. Calculate the limit and then see if it equals vervain. So we begin with Element A Z X approaches aim, and this case is too, of this entire quantity. Three acts of the fourth minus five x plus the cube root of the quantity X plus. For since we have three terms that are either being added or subjected, willies are some indifference. Longs to separate this and two three different limits. Luminous explosions to two of three x to the fourth power, minus the limited expertise to Dr Nix Place. The limit is six purchase, too, on the Q Bert, the cube root of ex squiggles for which I&#x27;m going to write as explosive squared plus four to the wanted power for the first two terms will be able to use our concept multipolar parliament, which means that we can pull out any constant, simple find limit further. They pull out a three from the first function and a fire from the second function here for into third limit, we use the power that&#x27;s st Stand. If a function is being raised to the power instead of element Reid, you may you may separate. You may raise the limit itself to that power and since we have two term that inside were we&#x27;LL also add in used the some alarm to separate this inner a function into to limits and then everything is raised to the one third power. Finally, we really with this limit three times X approaching to race with fourth power and it&#x27;s five times the limit is expertise to Neck of X is equal to close. The limit is exposed to X squared is two squared plus, the limited exposure is two of four is equal to four. This is always to the one there power appear three ten to the fourth. Power is going to be forty eight Minus. Frightened, too, is ten plus two. Squared applies for is eight, and the Cure eatery is crying, too. So fourteen minus tennis, thirty eight plus two is forty, and we see that free or tap directly plankton. Two friends. We will get the same result. F of two is exactly forty eight minus ten, plus the queue brutally, which is too. And that&#x27;s also forty. So we have prevented dysfunction is continuous city, um at a equals two for this function?

Leon Druch · Answer

Okay, we have a function G of x equals x minus 1/3. X plus six on the interval negative infinity to negative two. This is an open interval, not a closed interval at negative two. So that means the number negative two is not included in this interval. And that&#x27;s good because uh if X was allowed to equal negative to um then three times negative two would be negative six and then plus six would be zero and you can&#x27;t divide by zero But -2 is not in this interval. Uh so we do not have to worry about the denominator being zero anywhere on this interval. Now we are going to show uh that this function G of X is continuous on this interval. And uh for a function to be continuous, a function is continuous. G will be continuous at a some number A Okay, in this interval uh will show that it works for every number A indiscernible, but G is continuous at A If the limit of G of X as X approaches A is equal to G obey. This is what it means to have continuity. The limit of the function as X approaches a number A equals the value of that function at a. Yeah, let&#x27;s go ahead and over here to decide. Let&#x27;s find out what govs G of A. For any number A in this interval, Member, the only number in this is the only number that would give us trouble is negative two, but negative two is not in this interval. So for any number a G of a, simply plugged uh a in for XG of a would be a -1 Over three times a plus six. So that&#x27;s gff. Now, if we could show that the limit of G of X as X approaches a equals G of a, then we will have shown that she is continuous on the interval. So what we want to do now is find a limit of G fx, which is x minus 1/3 X plus six. Find a limit of this expression as X approaches A. Well, The limit of X -1 as X approaches a exists, it&#x27;s simply just a -1. And the limit of uh three times X plus six as X approaches A exists and is three times eight plus six. And so the limit of x minus 1/3, X plus six as X approaches A is going to equal to limit Of the Function and the Numerator X -1. As X approaches a divided by the limit of the function, the denominator, which is three X plus six as X approaches A. So in other words, the limit of a closure is equal to a close version of the limits. As long as the limit of the function in the denominator okay exists and is not zero Now, remember, the only number that would make three X plus six equal to zero is if X was equal to negative two and our number A is not going to be negative too because A is any number in this interval but negative two is not in this interval, so they will not be -2. And so we are allowed to take instead of the limit of the close ship. We can take the closure. And the fraction of the limits as long as the limit of the function that was in the denominator exists, which it will and is not zero, which of course won&#x27;t be zero. So, now, what are these limits? Well, the limit Of X -1 as X approaches a is simply just a -1. We can just substitute it directly in there. The limit of the function three times X plus six as X approaches A is going to be three times A plus six. So the limit of our function G f X as X approaches A equals a minus 1/3 8 plus six, but G of A also equals a minus 1/3, 8 plus six. So we just showed for any A. In this interval the limit of G of X as X approaches A is equal to gov. So the limit of G of X as X approaches A equals G of A. So G is continuous at A because the limit of G of X as X approaches A does equal Gov.

Julian Gerber · Answer

So we have the function h of X, and we want to determine whether H of X is continuous at X equals negative too. So first will find that h of negative two is defined. So h of negative two equals. We can just substitute negative to for all the exes. So in the numerator we have negative two plus three over negative two squared minus negative two or plus two minus one times negative, too squared, plus one that simplifies to negative two plus three. That&#x27;s one in the numerator and negative two squared. That&#x27;s four plus two minus one is five times five. That&#x27;s 1/25. So we know that age of negative two is defined. Now we have to find the limit of h of X as X approaches. Um, negative too. And because if you just plug in negative too, um, into age of X, the answer isn&#x27;t indeterminant. We don&#x27;t get 0/0 or, um, infinity over infinity. We just get a value 1/25. So we know that to find the limit, we can just do the same thing. So we get for the limits 1/25 so because each of negative two exists and it&#x27;s equal to the limit as X approaches. Negative two. We know that h of X is continuous, um continuous at X equals too negative, too.

Problem

Use the definition of continuity and the properti…

Problem 11 Easy Difficulty

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number.
$$f(x)=\frac{x \sqrt{x}}{(x-6)^{2}} \quad \text { at } x=36$$

Answer

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Continuity - Overview

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Thomas W. Hungerford, Douglas J. ShawContemporary Precalculus: A Graphing Approach

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Contemporary Precalculus: A Graphing Approach