Limit problem simplify the root

[StrSpeed]

Homework Statement

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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
(link to all my reductions)

 

[LCKurtz]

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

 

[StrSpeed]

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.

 

[pasmith]

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

 

[StrSpeed]

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

https://www.physicsforums.com/file:///C:/Users/Steven/Downloads/CodeCogsEqn.gif

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

 

[Noctisdark]

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

https://www.physicsforums.com/file:///C:/Users/Steven/Downloads/CodeCogsEqn.gif

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

 

[SammyS]

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.

 

[StrSpeed]

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!

 

[momoko]

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

 

[SammyS]

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.