Question

help on pre cal plzzz ill fan & medal

Answer 1

@whpalmer4

Answer 2

\[\lim_{n\rightarrow\infty} \frac{7n^4+1}{n^4} = \lim_{n\rightarrow\infty}\frac{7n^4}{n^4} + \lim_{n\rightarrow\infty}\frac{1}{n^4}\]

Can you evaluate those limits?

Answer 3

@Kelsey42 are you there? :-)

Answer 4

yeah sorry just saw you replied haah

Answer 5

limits really confuse me haha :/

Answer 6

i dont get what to do

Answer 7

well, I shall attempt to make it painless :-)

Answer 8

Do you have any intuitive understanding of what a limit is?

Answer 9

yea

Answer 10

sometimes I really wish OpenStudy had a voice option!

Good. Do you agree that the limit of a sum is the same as the sum of the limits of the individual pieces? For example,

\[\lim_{n\rightarrow 1} (n+n^2) = \lim_{n\rightarrow 1} n + \lim_{n\rightarrow 1}n^2\]

Answer 11

Okay yeah that makes sense

Answer 12

so, that's what I did to split that first limit into two more digestible chunks.

Let's look at the left hand limit first:

\[\lim_{n\rightarrow\infty} \frac{7n^4}{n^4}\]

What is the value of \[\frac{7n^4}{n^4}\]when \(n=1\)?

Answer 13

7

Answer 14

then + 1 right?

Answer 15

Good. what about \(n=2\)? \(n=3\)? \(n=2532342987234\)?

Answer 16

still 7, right? Is there ever going to be a case where it isn't 7? (other than \(n=0\), which isn't allowed because that would cause a division by 0)

Answer 17

7(2)^4/2^4 = 112/16 = 7 yess :)

Answer 18

we can simplify the fraction before we try to take the limit:

\[\lim_{n\rightarrow\infty} \frac{7n^4}{n^4} = \lim_{n\rightarrow\infty} \frac{7\cancel{n^4}}{\cancel{n^4} } = \lim_{n\rightarrow\infty} 7 = 7\]because the limit of a constant is just the value of the constant...

Answer 19

any questions about that?

Answer 20

nope

Answer 21

but where did the +1 go?

Answer 22

next to the 7n^4

Answer 23

oh, we're just doing the 1st of the two limits that I split the original limit into...the other one comes next :-)

Answer 24

good so far?

Answer 25

oh ok i thought we were done ;( haha yes

Answer 26

okay, now the other part, and then we'll add this part and the other part together to get our final result.

\[\lim_{n\rightarrow\infty} \frac{1}{n^4}\]you understand how I got that?
\[\frac{7n^4+1}{n^4} = \frac{7n^4}{n^4} + \frac{1}{n^4}\]

Okay, as \(n\rightarrow\infty\), what happens to the value of \(n^4\)?

Answer 27

an answer such as "it gets really big" is fine...

Answer 28

it becomes a smaller number

Answer 29

the value of \[\frac{1}{n^4}\]gets smaller, yes...but I was asking about just \(n^4\), which clearly gets larger, right?

Here's a table of the value of \(\large \frac{1}{n^4}\) for \(n=1\) to \(n=10\):

\[\begin{array}{ccc}
n & \frac{1}{n^4} & \text{decimal value} \\\hline\\

1 & 1 & 1. \\
2 & \frac{1}{16} & 0.0625 \\
3 & \frac{1}{81} & 0.0123457 \\
4 & \frac{1}{256} & 0.00390625 \\
5 & \frac{1}{625} & 0.0016 \\
6 & \frac{1}{1296} & 0.000771605 \\
7 & \frac{1}{2401} & 0.000416493 \\
8 & \frac{1}{4096} & 0.000244141 \\
9 & \frac{1}{6561} & 0.000152416 \\
10 & \frac{1}{10000} & 0.0001 \\
\end{array}\]
(I'm showing both the fraction and its decimal equivalent)

Answer 30

What do you think the ultimate limit of that is going to be as \(n\rightarrow\infty\)?

Answer 31

Would you say that for any value I wanted to choose, you could find a value of \(n\) that would make \[\frac{1}{n^4}\]smaller than my value?

Answer 32

any positive value I might choose, that is...

Answer 33

1

Answer 34

wait a minute. the value of \(\large\frac{1}{n^4}\) went from 1 to 0.0001 as \(n\) went from 1 to 10, and now you think that if we let \(n\rightarrow\infty\) the value of the expression will go to 1?

Answer 35

no haha

Answer 36

itll just keep getting smaller

Answer 37

i dont think there is a limit

Answer 38

What about 0? Can't we choose a value of \(n\) that will make \(\large\frac{1}{n^4}\) arbitrarily close to 0?

Answer 39

0?

Answer 40

or 1

Answer 41

No matter what value \(\epsilon\) you might choose for "how close to 0", I can pick a value of \(n\) which makes \(\large\frac{1}{n^4} < \epsilon\)

That means the limit is 0. There isn't a value where we actually get 0 by doing the division, but we can come arbitrarily close if we make \(n\) large enough.

If you do this on a calculator, punching in bigger and bigger numbers for \(n\), eventually the calculator will either report that the answer is 0, or give an error of some sort.

Answer 42

So our overall limit is this:

\[\lim_{n\rightarrow\infty}\frac{7n^4+1}{n^4} = \lim_{n\rightarrow\infty}\frac{7n^4}{n^4} + \lim_{n\rightarrow\infty}\frac{1}{n^4} = 7 + 0 = 7\]

Answer 43

ahhhh makes sense :) so it does come out to be 7 anyways

Answer 44

is the other one no limit?

Answer 45

From an intuitive point of view, if you stick in numbers for \(n\), the contribution of the "+1" in the numerator becomes very small as \(n\) gets bigger:

\[\begin{array}{cc}
\frac{7n^4+1}{n^4} & \text{decimal}\\\hline\\
8 & 8. \\
\frac{113}{16} & 7.0625 \\
\frac{568}{81} & 7.01235 \\
\frac{1793}{256} & 7.00391 \\
\frac{4376}{625} & 7.0016 \\
\frac{9073}{1296} & 7.00077 \\
\frac{16808}{2401} & 7.00042 \\
\frac{28673}{4096} & 7.00024 \\
\frac{45928}{6561} & 7.00015 \\
\frac{70001}{10000} & 7.0001 \\
\frac{102488}{14641} & 7.00007 \\
\frac{145153}{20736} & 7.00005 \\
\frac{199928}{28561} & 7.00004 \\
\frac{268913}{38416} & 7.00003 \\
\frac{354376}{50625} & 7.00002 \\
\end{array}\]

Answer 46

Pretty quickly it's just the equivalent of \[\frac{7n^4}{n^4}\]

Answer 47

yea so you could find the limit by just inputing each one into a calulator?

Answer 48

this is the same thing you do when evaluating the end behavior of rational expressions: you only look at the biggest exponent term in the numerator and the denominator, because the other terms don't have as much of an impact on the values

Well, you could guesstimate the limit...if you put \(n=100\) into the calculator you'd end up with \[\frac{700000001}{100000000}\]and your calculator would probably say that is 7...

It might not be so easy to recognize the value of a fraction in decimal form. For example, would you recognize 0.142857142 is 1/7?

Answer 49

no haha

Answer 50

So, simplify your fractions by splitting them apart and canceling common factors. Then, if the numerator has an exponent of the same degree as the denominator, the limit is just the ratio of their coefficients. If the numerator has an exponent of a smaller degree than the denominator, the limit is 0 (it will be the equivalent of our right hand limit), and if the numerator has an exponent of a larger degree, there will be no limit (because you'll be multiplying by some power of infinity)

Answer 51

\[\lim_{n\rightarrow\infty} \frac{7n^3}{n^4} = \]\[\lim_{n\rightarrow\infty}\frac{7n^3}{n^2} =\]

What are those two limits, respectively?

Answer 52

ah gotchaa

Answer 53

what do you mean respectively?

Answer 54

nvm sorry i read your question wrong

Answer 55

I gave you two problems, I'm just asking you to answer them in the order given so I know which answer goes with which problem :-)

Answer 56

First one the limit is 0
and second one there is no limit

Answer 57

is that right?

Answer 58

yeah, 0 for the first, and the second one goes to infinity

Answer 59

I'm a little unclear whether you should say the second one has no limit, or that the limit is infinity, come to think of it. Probably better to say the limit is infinity.

Answer 60

Your other problem was \[\lim_{n\rightarrow\infty} 2n^2-7\]

From what you know now, what is the limit of that bad boy?

Answer 61

i would think it has no limit

Answer 62

the limit is infinity...

Answer 63

yeah but the answer is there is no limit

Answer 64

no, the answer is infinity — there is a limit. it isn't a specific number, it's infinity.

http://www.wolframalpha.com/input/?i=limit+2n%5E2-7+as+n-%3Einfinity

Answer 65

an example of something where the limit does not exist is

http://www.wolframalpha.com/input/?i=limit+1%2Fn+as+n-%3E0

there's no limit there because if you approach 0 from the negative side, 1/n becomes a very large negative number, but if you approach from the positive side, it becomes a very large positive number.

Answer 66

Infinity is the same thing a no limit im pretty sure

Answer 67

no, it's not.

Answer 68

because i answered it as no limit and got it right

Answer 69

I agree that the problem may have been marked as correct when you said "no limit" for the answer. I do not agree that that is the correct answer for the problem, however.

Answer 70

oh yeah the examples you showed me make sense i get it

Answer 71

the answer for that one would be infinity becuase it does not approach negative infinity but if it did then it would be no limit huh?

Answer 72

Now, if you want to believe that all the mathematicians who built Mathematica and WolframAlpha got such a fundamental concept wrong, you are free to do so :-)

Also, the people who wrote the section titled "Near Infinities" on this page: https://en.wikipedia.org/wiki/List_of_limits must be included in the list of people who don't know what they are talking about, because they clearly give the limit of such an expression as infinity...

I'm unclear which problem you are referring to when you say "the answer for that one would be infinity"...

Answer 73

Time for me to attend to other matters, I hope I've helped strip away some of the mysteries of finding limits! :-)

Answer 74

Thanks for all the info & help you are great! :D