Limits...what am I doing wrong ? (Without hopital)

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In summary, the problem involves finding the limit as x approaches positive infinity of the expression sqrt(3x^5-2x) - sqrt(3x^5-7x). The attempt at a solution involved rationalizing and dividing by x^5, but it is unclear if this is the correct approach. Factoring out x^5 and bringing it out of the radical as x^5/2 may be a more appropriate solution. Dividing by and cancelling out x may also be a valid approach.
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AlexandraMarie112
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Homework Statement



Limx--->positive infinity sqrt(3x^5-2x) - sqrt(3x^5-7x)

Homework Equations

The Attempt at a Solution


The first thing I did was rationalize. I got (5x)/ [ sqrt(3x^5-2x) + sqrt(3x^5-7x)]. Then I divided the top and bottom by x^5 but this is where I feel like I am doing something wrong. But I don't know what. Am I even supposed to divide by x^5 ? Because the other problems I've solved so far have been with an x^2.
 
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AlexandraMarie112 said:

Homework Statement



Limx--->positive infinity sqrt(3x^5-2x) - sqrt(3x^5-7x)

Homework Equations

The Attempt at a Solution


The first thing I did was rationalize. I got (5x)/ [ sqrt(3x^5-2x) + sqrt(3x^5-7x)]. Then I divided the top and bottom by x^5 but this is where I feel like I am doing something wrong. But I don't know what. Am I even supposed to divide by x^5 ? Because the other problems I've solved so far have been with an x^2.
Factor x5 out of each radical, and bring each out of its radical as x5/2.
 
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What happens, if you simply divide by / cancel x?
 

Related to Limits...what am I doing wrong ? (Without hopital)

1. What is a limit?

A limit is a fundamental concept in calculus that describes the behavior of a function as it approaches a certain value. It represents the value that a function is approaching as its input approaches a given value.

2. How do I calculate a limit?

To calculate a limit, you need to evaluate the function at values that are closer and closer to the given value. This can be done by plugging in values on either side of the given value and observing the output. If the values get closer and closer to a specific number, then that number is the limit. However, if the values do not approach a specific number, the limit does not exist.

3. What is the difference between a one-sided and a two-sided limit?

A one-sided limit only considers the behavior of a function as the input approaches the given value from one side (either from the left or the right). A two-sided limit considers the behavior of the function as the input approaches the given value from both sides.

4. Can I use L'Hopital's rule to solve any limit?

No, L'Hopital's rule can only be used to solve certain types of limits, such as limits involving indeterminate forms (e.g. 0/0 or ∞/∞). It cannot be used to solve limits that do not have indeterminate forms.

5. What are some common mistakes when solving limits without L'Hopital's rule?

Some common mistakes when solving limits without L'Hopital's rule include overlooking the existence of a common factor that can be factored out, not considering the possibility of using algebraic manipulation to simplify the expression, and forgetting to check for the existence of a one-sided limit.

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