Limits: ay = 1/ ( y+ √ (y+ √ (y+ √ (y)))), ay = ( (3y-5) / (3y) ) ^(6y), ...

[Alpacca]

Limits: ay = 1/ ( y+ √ (y+ √ (y+ √ (y)))), ay = ( (3y-5) / (3y) ) ^(6y), ...

Hello!
I can`t solve this exercise and wanted to ask if someone might know the answer to it.
In need the limit to these orders:

ay = 1/ ( y+ √ (y+ √ (y+ √ (y))))

ay = ( (3y-5) / (3y) ) ^(6y)

ay = (√ (y + 1)) - √y

y is always a natural number (e.g 0,1,2,3,4....)

I am thankful for any answers or advices!

Ps I`m not a native speaker, so sorry for any grammar mistakes

 

[stapel]

In need the limit to these orders:

ay = 1/ ( y+ √ (y+ √ (y+ √ (y))))

ay = ( (3y-5) / (3y) ) ^(6y)

ay = (√ (y + 1)) - √y

y is always a natural number (e.g 0,1,2,3,4....)

Okay, but what is "a"? Are you finding limits, or solving equations? If finding limits, what are those limits? "The limit as [what?] tends toward [where?]" What do you mean by "orders"?

Thank you!

 

[Alpacca]

Hello!

"a" is just supossed to show that it`s an integer sequence (I looked it up again and that should be the correct english term... I used "order" because I didn`t know how to express this type of function)... And I need the limits from right and left from 0 to infinty/ - infinity.

 

[ksdhart2]

Alright, so if I'm understanding correctly, the problem statement says something like the following:

Consider the sequence defined by the general term:

\(\displaystyle \displaystyle a_y=\frac{1}{y+\sqrt{y+\sqrt{y+\sqrt{y}}}}\)

For this sequence, find:

i) \(\displaystyle \displaystyle \lim _{y\to 0^+}\left(a_y\right)\)

ii) \(\displaystyle \displaystyle \lim _{y\to 0^-}\left(a_y\right)\)

iii) \(\displaystyle \displaystyle \lim _{y\to \infty}\left(a_y\right)\)

iv) \(\displaystyle \displaystyle \lim _{y\to -\infty}\left(a_y\right)\)

And then similar for the other two sequences. If the above is not correct, please reply with any necessary corrections. In either case, please share with us any and all work you've done on these problems, even if you know it's wrong. The more specific you can be about exactly where you're getting stuck, the better help we can provide. Thank you.

 

[Alpacca]

The case you just wrote is correct!
Thank you for your help! I really appreciate it!

 

[ksdhart2]

I'll infer from the fact that you've not shown your work that you have none to show. That's okay - we'll start at the very beginning. Let's work with the first sequence and try to find the limit as y approaches 0 from above. We can tell just by plugging in 0 as the value for y, that the limit will be of the form 1/0. What do you know about limits of that form? Now, let's look just as the denominator. What do you notice about the values as y gets smaller and smaller? What does that tell you about the limit? If we consider the limit as y approaches 0 from below, what do you know about the domain of the square root function? What does that mean about the values of the denominator? What can you then say about the limit? Now let's consider the limit as y approaches +/- infinity. Based on what you saw when finding the limit as y approaches 0 from below, what can you automatically say about the limit as y approaches negative infinity? Now consider the values of the denominator as y grows without bound. What does that suggest about the limit as y approaches positive infinity?

For the second problem, the limits as y approaches 0 from either direction can be found by simply plugging in the value of 0. Note that anything raised to the power of 0 is 1. As for the limits at infinity, that's quite a bit more difficult, and I don't actually know how to get the answer. I'm hoping it will come to me as I continue to play with it, or perhaps another forum member here can give you some guidance.

The third problem has a defined value at the point y=0 so that takes care of those limits too. To find the limit as y approaches infinity (or negative infinity), you might try turning the expression into a fraction by multiplying by the conjugate and then use the same process you used for the first problem.