# Limit problem simplify the root

• StrSpeed
In summary, the person is having trouble solving a problem and has tried multiple methods with no success. They have come to the conclusion that they may need to use a constant or infinity in their equation in order to get it to reduce. They have found a way to do so and are grateful to their helpers.
StrSpeed

## Homework Statement

[/B]
I feel like I'm missing some theorem which is preventing me from finalizing this problem! It's been driving me nuts I feel like I'm missing something super basic!

Ultimately they've given the solution, g/8, so I know this is how I should try to get the equation to look algebraically. But, no matter how I manipulate it I can't get it to reduce.

I feel like this has to do with √x2 = |x| Which then depending on your value of x will give x, or -x. However, I can't simplify the root into a way which will let me make this jump.

## Homework Equations

https://www.desmos.com/calculator/hf8poewlvb

## The Attempt at a Solution

https://www.desmos.com/calculator/hf8poewlvb

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Try multiplying numerator and denominator by$$c\sqrt{\left(\frac{c^2}{g^2}+\frac 1 4\right)}+\frac{c^2}{g}$$

The conjugate! How could I forget that.. Thank you! Now I have:

PS. g is not squared I wrote it down wrong.

Let me see what more I can do.

Alternatively: You can take a common factor out of the expression under the square root to obtain $$h(c) = A\sqrt{1 + x} - \frac{c^2}{g}$$ where $x < 1$ for sufficiently large $c$. Hence you may expand the root as a binomial series, $$(1 + x)^{\alpha} = 1 + \alpha x + \frac{\alpha(\alpha - 1)}{2!}x^2 + \dots$$

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Thanks for both of your help! Still working on it, sadly.. I pulled out a C so I'm left with:

Im not entirely sure what I could pull out of the root.

Last edited by a moderator:
StrSpeed said:
Thanks for both of your help! Still working on it, sadly.. I pulled out a C so I'm left with:

Im not entirely sure what I could pull out of the root.
Factor out the c you'll have 1/4 * (lim c ->0 1/(sqrt(1/g^2 + 1/4c) + 1/g) and don't forget that 1/4c goes to 0 whenever c -> 0, good luck

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StrSpeed said:
The conjugate! How could I forget that.. Thank you! Now I have:

PS. g is not squared I wrote it down wrong.

Let me see what more I can do.
That's good.

##\displaystyle \ \frac14\lim_{c\to\infty}\left(\frac{c^2}{\displaystyle c\sqrt{\left(\frac{c^2}{g^2}+\frac 1 4\right)}+\frac{c^2}{g}}\right)\ ##

One way to deal with rational expressions where some factor →∞ : divide the numerator and denominator by the highest power of that factor.

Divide by c2 in the numerator & denominator.

I got it! You guys rock thank you so much! Once I got down to that last x Term I realized that a constant/infinity = 0!

Here is my work through!

You should drop the limit sign from the second last row.

StrSpeed said:
I got it! You guys rock thank you so much! Once I got down to that last x Term I realized that a constant/infinity = 0!

Here is my work through!
At the point highlighted above, simply take the limit and simplify.

## 1. What is a limit problem?

A limit problem is a mathematical concept that involves finding the value of a function as the input approaches a certain value. It is used to determine the behavior of a function at a specific point or when the input approaches infinity.

## 2. How do you solve a limit problem?

To solve a limit problem, you need to follow a set of steps. First, you need to evaluate the function at the given input. Then, you need to check if the function is continuous at that point. If it is, you can simply substitute the input value and get the limit. If not, you need to use algebraic manipulation or other techniques, such as L'Hopital's rule, to simplify the function and find the limit.

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

A one-sided limit only considers the behavior of a function as the input approaches a certain value from one direction, either from the left or from the right. A two-sided limit, on the other hand, considers the behavior of a function as the input approaches a certain value from both directions. It is possible for these two types of limits to have different values.

## 4. Can you use a graph to solve a limit problem?

Yes, a graph can be a helpful tool in solving a limit problem. By looking at the graph, you can visually determine the behavior of the function at a certain point or as the input approaches infinity. This can give you a better understanding of the limit and help you find the correct solution.

## 5. How do you simplify the root in a limit problem?

To simplify the root in a limit problem, you can use various techniques such as factoring, rationalizing the denominator, or using trigonometric identities. The goal is to transform the function into a simpler form that allows you to evaluate the limit. It is important to carefully follow the rules of simplifying roots to avoid making mistakes in the calculation.

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