lim_x->inf((x+2)/sqrt(9x^2+1))

[ahorn]

Here is the question I'm working on:

lim_x->inf ( (x+2) / sqrt(9x^2 +1) )

I have tried squaring ( (x+2) / sqrt(9x^2 + 1) ) and putting that in a +-sqrt. Then I took the lim to the inside of the +-sqrt and solved the limit to be 1/9. So, my anwswer is +-sqrt(1/9) = +-1/3. How do I find the correct solution out of those two numbers?

 

[pka]

Here is the question I'm working on:

lim_x->inf ( (x+2) / sqrt(9x^2 +1) )

I have tried squaring ( (x+2) / sqrt(9x^2 + 1) ) and putting that in a +-sqrt. Then I took the lim to the inside of the +-sqrt and solved the limit to be 1/9. So, my anwswer is +-sqrt(1/9) = +-1/3. How do I find the correct solution out of those two numbers?

Note that \(\displaystyle x>0\) and \(\displaystyle \dfrac{\sqrt{9x^2+1}}{x}=\sqrt{9+\dfrac{1}{x^2}}\)

 

[Deleted member 4993]

Here is the question I'm working on:

lim_x->inf ( (x+2) / sqrt(9x^2 +1) )

I have tried squaring ( (x+2) / sqrt(9x^2 + 1) ) and putting that in a +-sqrt. Then I took the lim to the inside of the +-sqrt and solved the limit to be 1/9. So, my anwswer is +-sqrt(1/9) = +-1/3. How do I find the correct solution out of those two numbers?

Did you look at the graph of \(\displaystyle y \ = \ \dfrac{x+2}{\sqrt{9x^2+1}}\)?

 

[Quaid]

ֺ

lim_x->inf ( (x+2) / sqrt(9x^2 +1) )

I have tried squaring ( (x+2) / sqrt(9x^2 + 1) ) and putting that in a +-sqrt.

my anwswer is +-sqrt(1/9) = +-1/3. How do I find the correct solution out of those two numbers?

Hi ahorn:

Both the numerator and the denominator are heading toward large, positive values. There are no negative values, in the given expression. There are no subtractions, in the given expression. Therefore, there is no way for this ratio of positive values to head toward -1/3 as x heads toward positive infinity. Make sense?

Here's a different approach (along the lines of what pka is hinting at).

Firstly, whenever we don't have information about the sign of x, we must say that √(x^2) = |x| = ±x

But, in this limit exercise, we know that x is always positive, so we can say that √(x^2) = x

Secondly, we can factor 9x^2 out of the radicand.

√(9x^2 + 1) = √(9x^2*(1 + 1/[9x^2]))

Knowing that √(x^2) = x, can you simplify this radical, in the denominator?

Thirdly, we can divide top and bottom by x (because we know that x is never zero, here).

Lastly, you probably already know that expressions such as 1/(9x^2) and 2/x head toward zero, as x becomes super large.

What do you think of this approach?

Cheers

ֺ

 

[ahorn]

Thank you all for your thoughts. Pka's method is useful (I was taught how to take the x^2 out of a radical a while ago, but I forgot it). I can see how a positive x will result in a positive number.