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[6 points] List the conditions for f (x) to be continuous at x = a[6 points] Use the conditions to check following statements true or false: If a function is contin...

Question

[6 points] List the conditions for f (x) to be continuous at x = a[6 points] Use the conditions to check following statements true or false: If a function is continuous at x =a then If a function does not have a limit at has a limit at 1=0 x=a,itis still possible to have left or/and limits at x =aIf a function is discontinuous atx = a then it is undefined at x = aIf a function is undefined at x = 4 , then it does not have limit at x =a.If a function is continuous at x = 4, then If a functions ha

[6 points] List the conditions for f (x) to be continuous at x = a [6 points] Use the conditions to check following statements true or false: If a function is continuous at x =a then If a function does not have a limit at has a limit at 1=0 x=a,itis still possible to have left or/and limits at x =a If a function is discontinuous atx = a then it is undefined at x = a If a function is undefined at x = 4 , then it does not have limit at x =a. If a function is continuous at x = 4, then If a functions has a limit at x = a , then it it is both left continuous and right is continuous at x =a as wcll continuous as well:



Answers

Classify each statement as either true or false.
If $\lim _{x \rightarrow 4} F(x)$ exists, then $F$ must be continuous at $x=4$.

In discussion or in this problem, we have to decide whether or not the function is continuous. If it is not continuous, we will. We have to identify the points at which it is discontinuous, so the human function is F of X is equal to zero. There's a piece wise function if X is less than zero, and eggs if X is greater than are equal to zero, is the value of the function. F of X is difficult. F of X is equal to zero for when X is less than zero. It means that this function is continuous for all the values of X, which is less than or equal to zero. Similarly F of X is equal to the value of the function is X when X is greater than are equal to zero again. F of X is equal to X is a polynomial, so this will be continuous for all the values of X greater than zero. Now we will check. The continuity of the function at X is equal to Europe, so it for continuity for a function to be continuous, it must satisfy the conditions. The three conditions of the of the function. Yeah, the three conditions that are the the function should be defined at the given point. So f of zero is equal to f of the urological too. We have a pope accessible tracks for X greater than or equal to zero. So f of zero is equal to zero for X is equal to zero. So the given function is define at X is equal to zero. Now we will check the We will check whether the limit exists at X radical 20 or not. So, first of all, we will find the left and side limit, So limit X approaches to zero from left hand side of the function as X is less than zero because X approaches to zero from left hand side, it means that access less than zero. So from the function, the value of the function is zero when X is less than zero. So we have zero now the right hand side limit. Now we will try to find the right hand side limit, so limit X approaches to zero from right hand side. Therefore, effects as X approaches to zero from right hand side. It means that X is greater than zero. So from that function we have, the value of the function is X. When X is less, the X is greater than zero. So this implies that we have on applying limit. We have zero again, so left hand side limit is equal to right hand side limit. So this implies that limit X approaches to zero f of X is exist. And finally the third condition is this is the second condition, and the third condition for the continuities limit of the function at the given point should be equal to the value of the function at that point. So limit X, brought just during f of X is equal to zero because left hand side limit and right hand side limit is equal. So we have lim X approaches to zero f of X is equal to zero and the value of the function at X is equal to zero is also zero. Yeah, so all the properties of the continuity is satisfied by the human function. So we can say that the human function is continuous everywhere

So in this question were given, the problem is for his equation for f of X, which is X squared minus 36 Over X -6, Even X is not equal to six and if X is a call to sex It's going to be 13, so 13, if X is equal to six having this, we need to see if the function is continuous at a is equal to six. So in order to figure out continuing continually, we need to show that these values are equal and if of six is a call to the limit as X goes to six from the left of F a bex and also equal to the limit as X goes to six from positive over for Becks. Now, if f of we have about six then As um what is FSX is 13, we have it here. So therefore to here we have 13 for the limit as X goes to sex from the negative of f. Well, let's just rewrite it here. That is going to be the limit as X close to six from negative of X word minus 36 Or X -6, that's the definition of that. Uh active coaches checks from the left now, that is going to be the same as if I simplify it, the numerator is x minus x times X plus six And after it simplification and just get an X-plus six As X costs six. This limit becomes 12. So we already see, we don't have any equality between the first two terms. So we already know that our function is not continues at six.

Hello. So here are given function is F of X is equal to one over X now. Um If we look at the function F of negative acts so F of negative X is going to be equal to negative one over X. Um shows that for positive values of X, the function is gonna provide positive results in the first quadrant. And also then for negative values of X, the function is going to give us positive results in the third quadrant. So the function um is going to be symmetric about the origin. Now for the function F of X equals one over X at X is equal to zero. There is going to be a discontinuity in the graph of the function. Therefore the statement that the graph defined by F of X equals one over X is continuous is false.

Yeah. In this problem, we have to decide whether or not the function is continuous. If it is not continuous, identify the point at which it is discontinuous. So the human function is f of X is equal to X. You access less than five and we have a piecewise function. Extra care Ive X is greater than are equal to zero is greater than or equal to five. We have to check the continuity of the function and we have to identify the points and with this function is not continuous. So this is a piecewise function as F of X is equal to X. If X is less than five. So f of X is equal to X is a polynomial function. So we know that every polynomial function is continuous everywhere. So in this particular case, access less than five. So for all the values of X, less than five dysfunction is defined so we can say that it's got This function is continuous because fo fact is a polynomial function. Similarly f of X is equal to f of X is equal to X care for X is greater than five. It's again a polynomial function So this function will also be continuous or X is greater than five Now we will take now we will take whether the given function is continuous and accessible to fight or not. So, using the properties of continuity, let us check one by one. The first can first property of the continuities. The function should be defined at that point, so f of X is equal to extra care. For this is the function for X is equal to five. So when we have when we plug X is equal to fight in the function we have 25. So this function is defined at exit equal to five. This is the first condition for a function to re continuous at a point that is, the function should be defined at that point. Now the second condition is we will check the limits. We will check the limit is the limit exists at X is equal to five or not. So for that we have to find the left hand limit and right hand limit. First of all, we will find the left hand limit so limit X approaches to five. Yes, Airport X is equal to S X approaches to fight from left hand side. So we will take this part of the function for access less than why we have X. So when we apply the limit, we have right and for right hand side limit limit X approaches to five from right hand side. Airport X is equal to S X is greater than five. So we take ex occurred from the given function because the value of the function is executed when X is greater than or equal to fight. So we have extra care. So on your on while reading the limit, we have the value 25. As you can see, that left hand side limit is not equal to right and side limit, so we can conclude that limit ex abroad just to five f of X does not exist means that at X is equal to five. Left hand side limit is not able to right hand side limit, so Bellamy does not exist at access equal to five, so the second condition is failed. So the given function does not satisfy the second condition for a function to be continuous. So we can say that the given function is discontinuous at X is equal to five except accessible to five. The human function is continuous everywhere


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