How would I solve this limit?

  • Thread starter holezch
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  • #1
251
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Homework Statement



(2+y^2)/sqrt(4+y^4) as y goes to positive infinity




Homework Equations





The Attempt at a Solution



I divided it out by y^2 to get

lim y-> inf+ (2+y^2)/y^2 = 1 and the bottom lim y -> inf+ sqrt(4+y^4)/y^2 I can't solve..

by the way, I can't use L'hopitals rule

thanks!!
 

Answers and Replies

  • #2
128
0
Split them into 2.

1. [tex] 2 + y^2[/tex] --> inf as y --> inf
2. [tex] 1/sqrt(4 + y^4) [/tex] --> 0 as y -- inf

So you are having an inf / 0 situation.
 
  • #3
251
0
Split them into 2.

1. [tex] 2 + y^2[/tex] --> inf as y --> inf
2. [tex] 1/sqrt(4 + y^4) [/tex] --> 0 as y -- inf

So you are having an inf / 0 situation.


I know, that's why divided it out by y^2.. so the question really comes down to solving

lim y -> inf+ sqrt(4+y^4)/y^2
thanks for reading
 
  • #4
73
1
First of all, notice that (2+y^2)^2=(4+4y^2+y^4)>(4+y^4) for y positive.
This means (2+y^2)>sqrt(4+y^4)... which means that the limit you're looking for is either greater than or equal to 1 or infinite (why?).

As for evaluating the actual limit, you might find it easier to first deal with the ratio (4+4y^2+y^4)/(4+y^4)
 
  • #5
35,129
6,876

Homework Statement



(2+y^2)/sqrt(4+y^4) as y goes to positive infinity




Homework Equations





The Attempt at a Solution



I divided it out by y^2 to get

lim y-> inf+ (2+y^2)/y^2 = 1 and the bottom lim y -> inf+ sqrt(4+y^4)/y^2 I can't solve..

by the way, I can't use L'hopitals rule

thanks!!
Factor y^2 out of both the numerator and denominator and then take the limit.
[tex]\frac{2 + y^2}{\sqrt{4 + y^4}}~=~\frac{y^2(2/y^2 + 1)}{y^2\sqrt{4/y^4 + 1}}[/tex]

Can you finish it?
 
  • #6
251
0
Factor y^2 out of both the numerator and denominator and then take the limit.
[tex]\frac{2 + y^2}{\sqrt{4 + y^4}}~=~\frac{y^2(2/y^2 + 1)}{y^2\sqrt{4/y^4 + 1}}[/tex]

Can you finish it?

thanks, I can see that for large y, it is 1
 
  • #7
35,129
6,876
Yes. And you can see that in the original expression, as well. For large y, the numerator is pretty close to y^2, and the denominator is also pretty close to y^2, since the square root of y^4 is y^2. The constants in the top and bottom are less and less significant the larger y gets.
 

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