## Help needed to evaluate limits

Limits, differentiation, related rates, integration, trig integrals, etc.
mathjoy
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### Help needed to evaluate limits

Hello,
I have a problem to evaluate these
a) lim (as x goes to infinity) of (square root of [x^2+7+1]-x)

b) lim (as x goes to - infinity) of (square root of [x^2+7+1]-x)
I have no idea how to solve it.

Update!
I just did this question:
a) 3.5
B) infinity, but I am not sure why positive infinity??? Can someone explain it to me, please?
The answers are right, I just need to find out why for part b) positive infinity

anonmeans
Posts: 84
Joined: Sat Jan 24, 2009 7:18 pm
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### Re: Help needed to evaluate limits

I have no idea how to solve it.

Update!
I just did this question:
a) 3.5
B) infinity, but I am not sure why positive infinity??? Can someone explain it to me, please?
The answers are right, I just need to find out why...
How did you "do" these if you don't know how they're done?
a) lim (as x goes to infinity) of (square root of [x^2+7+1]-x)
(I'm going to assume that the "7" is supposed to be a "7x".)

Right now, the expression is "infty - infty = 0" because sqrt{x^2} = x and the 7x and the 1 don't really matter. But that's "indeterminate". The regular way to get around this problem with addition or subtraction with roots is to do the conjugate:

$\left(\dfrac{\sqrt{x^2\, +\, 7x\, +\, 1\,}\, -\, x}{1}\right)\, \left(\dfrac{\sqrt{x^2\, +\, 7x\, +\, 1\,}\, +\, x}{\sqrt{x^2\, +\, 7x\, +\, 1\,}\, +\, x}\right)\,$

$=\, \dfrac{(x^2\, +\, 7x\, +\, 1)\, -\, x^2}{\sqrt{x^2\, +\, 7x\, +\, 1\,}\, +\, x}\, =\, \dfrac{7x\, +\, 1}{\sqrt{x^2\, +\, 7x\, +\, 1\,}\, +\, x}$

The bottom doesn't divide by zero and the top isn't "0 - 0" anymore so you can take the limit. One way is to divide everything by x:

$\dfrac{\left(\dfrac{7x}{x}\, +\, \dfrac{1}{x}\right)}{\left(\sqrt{\dfrac{x^2}{x^2}\, +\, \dfrac{7x}{x^2}\, +\, \dfrac{1}{x^2}\,}\, +\, \dfrac{x}{x}\right)}\, =\, \dfrac{\left(7\, +\, \dfrac{1}{x}\right)}{\left(\sqrt{1\, +\, \dfrac{7}{x}\, +\, \dfrac{1}{x^2}\,}\, +\, 1\right)}$
b) lim (as x goes to - infinity) of (square root of [x^2+7+1]-x)
This time (because x is negative) you have to do the absolute value part: sqrt{x^2} = |x|. Since x < 0 then |x| = -x. So you have "-x - x = -2x". Evaluate.

mathjoy
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### Re: Help needed to evaluate limits

Anonmeans, thank you, but can you please answer my question?? Here it is again, in part b) why it is positive infinity? I have the answer 7/0, but if it goes to - infinity isn't it suppose to be negative?

As for part a) I rationalized the expression (I am not sure how to use the tool on this site to show the calculations, I can type with words, but it would be very confusing) Here is the final answer I got for part a) after rationalizing 7/(square root 1) +1=7/2=3.5
Can someone help with part b)?

anonmeans
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### Re: Help needed to evaluate limits

I gave you all the steps. I showed you the limit expression is -2x. I pointed out that x < 0. Algebra says x < 0 means -2x > 0 so, when you take the limit, the value has to be positive. What part do you still need answered for you?

mathjoy
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### Re: Help needed to evaluate limits

I gave you all the steps. I showed you the limit expression is -2x. I pointed out that x < 0. Algebra says x < 0 means -2x > 0 so, when you take the limit, the value has to be positive. What part do you still need answered for you?
Do you know how to read?? I asked this question twice already, if you don't have the answer, please do not reply. I didn't ask you to list "all the steps". By the way your steps are not correct, I checked with professor and you are wrong. This is not productive at all.

I solved my own question, so there is no need to reply, thanks.

anonmeans
Posts: 84
Joined: Sat Jan 24, 2009 7:18 pm
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### Re: Help needed to evaluate limits

Do you know how to read?
Yes, I do. Do you?

I showed you all the steps and explained that the result means that the limit of negative 2 times negative numbers, as those numbers get really, really far negative, is positive infinity, so the limit is "+infty". Why do you think showing your work is not part of finding the answer? Which part of complete worked solution do you still think is missing? Why do you think my giving you the whole solution means that I "don't have the answer"?
By the way your steps are not correct, I checked with professor and you are wrong.
Really? I'd love to see any evidence for that.
I solved my own question
Great. Show me your working. Maybe you can explain it to me, since I'm so stupid.