Solving a Limit Without L'Hopital's Rule

  • Thread starter holezch
  • Start date
  • Tags
    Limit
In summary, to identify the type of limit, you should check for patterns and refer to a table or guide. Solving a limit algebraically involves simplifying and manipulating the expression. An indeterminate limit results in an undefined or infinite value and may require advanced techniques. A graph or table can be used to estimate a limit, but it should not be the sole method. Common mistakes to avoid include not checking for existence, incorrect algebraic manipulations, and not simplifying fully. It is also important to consider restrictions and the function's behavior at the given value.
  • #1
holezch
251
0

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!
 
Physics news on Phys.org
  • #2
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
Unto said:
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
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
holezch said:

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
Mark44 said:
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
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.
 

FAQ: Solving a Limit Without L'Hopital's Rule

How do I identify the type of limit and which method should I use to solve it?

To identify the type of limit, you should first check for any common patterns or algebraic expressions that may indicate a limit of a specific type (e.g. polynomial, rational, exponential, etc.). Then, you can refer to a table or guide that lists the different types of limits and the corresponding methods to solve them.

What is the process for solving a limit algebraically?

The general process for solving a limit algebraically involves simplifying the expression as much as possible, plugging in the given value of the variable, and manipulating the expression using algebraic properties and rules until a value for the limit is obtained.

How do I know if a limit is indeterminate and how should I approach solving it?

A limit is considered indeterminate if it results in an undefined or infinite value. You can determine if a limit is indeterminate by plugging in the given value and evaluating the expression. To solve an indeterminate limit, you may need to use more advanced techniques such as L'Hopital's rule or factoring.

Can I use a graph or table to solve a limit?

Yes, a graph or table can be used to estimate the value of a limit, but it should not be used as the sole method of solving the limit. You should always try to solve a limit algebraically first before using visual aids.

What are the common mistakes to avoid when solving a limit?

Some common mistakes to avoid when solving a limit include forgetting to check for the existence of a limit, using incorrect algebraic manipulations, and not simplifying the expression fully before plugging in the given value. It is also important to keep track of any restrictions on the variable and to consider the behavior of the function at the given value.

Similar threads

##\lim_{x \to0} \left(\dfrac{1}{\sin^2 x}-\dfrac{1}{x^2}\right)##
Replies
5
Views
147
Limit problem involving two circles and a line
Replies
8
Views
1K
Electromagnetism: Force on a parabolic wire in uniform magnetic field
Replies
20
Views
1K
How to solve a limit in two variables with an indeterminate form at (0,0)?
2
Replies
35
Views
3K
L'Hopital's Rule case: How does x^(-4/3) equal 0 when x approches infinity?
Replies
1
Views
832
Question re: Limits of Integration in Cylindrical Shell Equation
Replies
24
Views
2K
Relationship between autonomous system and related single equation
Replies
3
Views
1K
What is my mistake in solving for this limit?
Replies
23
Views
2K
Limit when x^2 + y^2 -> inf, am I solving it correctly?
Replies
8
Views
1K
Proofs about the second-order linear differential equation?
Replies
21
Views
2K

Hot Threads

Recent Insights

Back
Top