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7. Using one of the three conditions in me definition of cantinuity givemneknearoahei leachs7gconois discontinuous aoR} Meiptlea)IBONUSI Sketch function . {(xllyuch...

Question

7. Using one of the three conditions in me definition of cantinuity givemneknearoahei leachs7gconois discontinuous aoR} Meiptlea)IBONUSI Sketch function . {(xllyuch that 0-4) 5 and lim f (r) =0

7. Using one of the three conditions in me definition of cantinuity givemneknearoahei leachs7gconois discontinuous aoR} Meiptlea) IBONUSI Sketch function . {(xllyuch that 0-4) 5 and lim f (r) =0



Answers

Find each function value and limit. Use $-\infty$ or $\infty$ where a ppropriate. $f(x)=\frac{5 x+11}{7 x^{3}-2}$ (A) $f(8)$ (B) $f(16)$ (C) $\lim _{x \rightarrow-\infty} f(x)$

So we have this function. And the first thing we're gonna do is just find what this functions value is equal to at eight. So we're looking at F of eight. So we plug in eight into our function five times eight is 40 plus 11. And then we're dividing by eight cubed And I'm just plugging this into my calculator. So cubed multiplied by seven, It's equal to 3584 -2. So this is equal to 51, divided by 3582. Mhm. And that is equal to if I plug it into my calculator about .014, let's now look at what F. Of 16 is equal to, So 16 times five is going to be equal to 80 and then we have plus 11 and we're dividing by um 16 to 30 or 16 cubed, multiplied by seven Which is equal to 28,670- -2. And so this is equal to 91 divided by 28,670. And that value is equal to approximately point oh three. And then the last thing we're gonna do is just look at the limit as our function goes towards negative infinity. So what does this function do as we go towards negative infinity? Well, let's look at what our function was equal to. Again, it's five X plus 11 divided by seven X cubed minus two. So we have a polynomial divided by a polynomial. So this is going to be equal to the limit as X goes to negative infinity of the leading term of a polynomial in the numerator which is five X divided by the leading term of the polynomial and our denominator Which is seven x. cubed. And now we can simplify this by cancelling out one X term. And so this is going to be equal to the limit as X goes to negative infinity of five divided by seven X squared. And we can see that our numerator is a constant in our denominator is increasing towards infinity, so we're gonna have five divided by infinity, Which is equal to zero.

The function it is given that five X plus 11 up on seven x two minutes to seven x few minutes to Yeah. In the party, we have to find half of minus eight. So it will leave five to minus eight. Just 11. Seven to minus eight to the power of three. Manage to cause too. Five into minus eight. Last 11, about seven into minus eight to the power of three. Managed to Yeah, so it will be passed to 0.0 zero 12008 08 Okay, well, we can right one double zero. Anyone for the party will be able to find have four months. 16, That is five into minus 16. Plus 11 upon five and 2000 and 16 to the power of three months or two. So it will be close to like this. $1. 0. So four 11024 What? Yeah. This is approximately one to was beautiful. So we have to find the limited part. C limit extends to infinity minus of NVIDIA FX. FX has given five X plus 11 upon 71 ST. It will be cost limit exchange to minors of Infinity X will take common. We will take X comin from the narrator and xq from the denominator. Seven months go by, experts. So it will be a close to zero when we put elements that will be close to zero. Because when my ex one by minus NVIDIA is zero. I hope I answered your affection for the part of the answer for the party, Mrs. Thank you. Yeah.

So we have this function F. X. Were first just going to find the value of F. Of negative six. So we're just gonna plug in negative six into our function. So we have seven times negative 6 -14 times negative six squared, divided by six times negative six to the fifth plus three. And I forgot to put my four here and I forgot to put my squared here. Um And so what we're gonna do is I'm just gonna plug these values into my calculator. So seven X. seven times negative 6 to the 4th. It's going to be equal to 9072. We have 14 times X. or 30 negative six squared which is equal to 36. So this is gonna be equal to 504. And then we have divided by six times negative six to the fifth. So That's going to be equal to negative 46,000 606 and 56. And then we have plus three. So 9070 to minus 504 is equal to 8,568. And then we have divided by negative 4th, 46,000 653. And so this is the value of F of -6. Let's now look at f of negative 12. And so we're plugging in negative 12 into our function. And so we're going to have seven Times -12 to the 4th minus 14 times negative 12 squared, divided by six times negative 12 to the 5th plus three. And so I'm gonna plug this into my calculator, negative 12 to the fourth, multiplied by seven is equal to 145,000 152. And then this is minus 14 times negative 12 squared, which is equal to 2016. And then we have divided by -12 to the 5th Um multiplied by six And then we add three and that's equal to one million. So it's equal to negative one million. 400 In 92,000 989 And 145,150 to minus 2016. It's equal to. So this is equal to -143000 136, divided by 1,000,492,000 989. And so if I plug this into my calculator this is going to be equal to um point- .09 59 Mhm. And then the last thing we're gonna do is we're just going to look at the limit as our function goes towards negative infinity. So our function was seven X to the fourth minus 14 X squared divided by six X to the fifth Plus three. And so we have a polynomial divided by a polynomial. And therefore this limit is only going to be determined by the polynomial in the numerator term with the biggest degree divided by the polynomial and the denominators term um with the biggest degree. So we only look at the leading term of the term with the biggest degree in the numerator and the denominator. And so this is going to be equal to the limit as X goes to negative Infinity of seven x to the 4th divided by six X to the 5th. And we can reduce this or simplify this by cancelling out some X terms. And we're going to get this is equal to seven divided by six X. And now we can see and this should have been negative infinity. Now we can see that our numerator is just equal to a constant and our denominator is going to be growing towards negative infinity, so this is going to be equal to zero since we're going to have a constant divided by negative infinity, which is equal to approximately zero.

Okay so we have this function F. Of X. First thing we're gonna do is we're gonna find F. Of five. So we plug in five for X. So this is equal to two times 3 Times 5 to the 3rd, Divided by seven plus 4 times five to the third. And five to the third is equal to 1 25. And 1 25 times three is equal to 3 75. So this is gonna be equal to 2 -3, divided by seven plus four times 1 25 which is equal to 500. So this can be equal to seven plus 500. And so this is equal to 2 -375 is equal to negative 373 divided by 507. We're now going to find the value of f. of 10. So you plug in 10 for X. We're gonna get to minus three times 10 to the third, divided by seven Plus four times 10 to the third and 10 to the third is equal to 1000. So this can be equal to two minus 3000, Divided by seven plus 4000 which is equal to -2998 Divided by 4007. So we can see that these values are going towards negative 3/4 since negative 2998 divided by 4000 and seven. It's very close to negative 3000, divided by 4000, which is equal to negative 3/4. And now we're just going to look at the limit as X goes to infinity of our function which was two minus three X to the third, divided by seven plus four X to the third. So whenever we have a limit as X goes to infinity of a polynomial like this, of a of a polynomial divided by another polynomial. That means that this is only going to depend on the leading term or the term with the highest degree in the numerator in the denominator. So this is going to be equal to the limit as X goes to infinity of negative three X to the third, divided by four X to the third, and now we can get rid of these X to the third terms. And we're left with this equaling just negative 3/4. So we did now confirm that the limit as X goes to infinity is equal to negative 3/4.


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