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Question Completion Status:QUESTION 16Provide an appropriate response.continuous at x = 4222<Xso Sx<2 Sx $4 7 = 4E(x) =Click Save and Submit t0 save and submit. Click Save All Answers iae all answersSave Al Answe
Question Completion Status: QUESTION 16 Provide an appropriate response. continuous at x = 42 22<Xso Sx<2 Sx $4 7 = 4 E(x) = Click Save and Submit t0 save and submit. Click Save All Answers iae all answers Save Al Answe
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In Problems $73-76$, discuss whether $R$ is continuous at $c$. Use limits to analyze the graph of $R$ at $c$. Graph $R$. $$R(x)=\frac{x^{2}+4 x}{x^{2}-16}, \quad c=-4 \text { and } c=4$$
Does a given problem, we want to determine um where the intervals in which F of G is continuous. So we know that F of X is equal to one over root X -1, and G fx is exports for. Yeah, so our composition, which we call H of X An equal one over the square root of X plus four minus one, which is really just gonna be X plus three. So we know two things that's down here in the fraction we cannot have a zero, so X can't be a negative free. And then in the radical we can't have a negative value. So this can't be negative, meaning X can't be less than negative three. So because of that, we know that the biggest that X can be X has to be greater than or greater than -3. So because of that, that's going to be our interval of continuity. Um from negative three to infinity be our final answer.
Okay, so we wanted to find Chief Foreign away. That extends its function. Energy of Exodus continues the Mexicans for Okay, So the function is undefined it for So we would like to just send Jia for to be equal to the limit His exit purchase for the dysfunction X squared minus sixteen over X squared minus three X minus four You can every factor this limits except purchase for the top is the difference of squares X minus four times X plus for it and the bottom factors as Tex minus four Expo's. For those are X plus one go in the X minus force will cancel. You are left with Lim is experts for of X plus four over experts, one which is a proper five.
So with them, we're doing the same thing. Ah, new mayor. Can you be written as X minus for right X plus four? Because this 16 is sumas four squared, right? And the denominator. Should you have to factor in the determine before this one? Right? Eso This one is also and you do the same. Process X minus four experts one rate and this one is canceling this one. So you haven't, uh, excellence for X plus one, right? So now if you find g four e going to find you for us, an extension off just ah g a. Thanks. This is gonna be for us for over four plus one, which is 8/5. Right now we take the limits because we really can't new any. Right? So we take the limit as X approaches. Four off this G effects this extension. Ah, and that is also for plus for over four plus five is eight 05 Right. So limits and the functional value are the same. Therefore, uh, for extension, Geo four is is continuous. Adds eggs equals four
To determine whether the piecewise function is continuous at sea equals zero. We begin by checking the conditions for continuity. Now the first one would be to check if F of zero is defined. Now since X equals zero is in the domain of the function, then F of zero is defined And this is equal to four. And then the next one would be if the limit of dysfunction As X approaches zero exists. And now because we're dealing with piecewise function, then you have to find the one sided limits. First. Want to find if the limit of the function As X approaches zero from the right Equals the limit of the function as X approaches zero from the left. So let's find limit of the function as X approaches zero from the left 1st. So as approaching zero from the left, that means we're looking at That function whose X values are less than zero. And so we're gonna use 4 -3 x squared. And this will be limit as X approaches zero from the left of 4 -3 x squared. And this is just four minus three times zero square or four. And then for the limit of the function As X approaches zero from the right, we're going to use that function whose X values are greater than zero. And in this case we're going to use the radical. So we have limit as X approaches zero from the right of the square root of 16 minus X squared over four minus six. And here we have The squares of 16 0 over Format zero. This is just The square at the four, which is equal to two. And because these values are not equal, then you say that the limit does not exist unless you want to check if the limit of this function As X approaches zero equals the value of the function at zero. Because the limit does not exist, then this will never be equal to The value of the function at zero. So this is not true, and we say that we can. The unity of the function is not told that C equals zero, or this is discontinuous.