Limit of a function

[Albi]

Guys I'm trying to solve this problem but I'm having difficulties, here is what I tried:
$$\lim_{{x}\to 0} \frac{\sqrt[n]{c+x} - \sqrt[n]{c-x}}{x}, c>0, n\in \mathbb{N}$$

$$\lim_{{x}\to 0} \frac{\sqrt[n]{c+x} - \sqrt[n]{c-x}}{x} \times \frac{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}}{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}} = \lim_{{x}\to0}\frac{2}{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}}$$
how should I continue or should I use another approach?

 

[BigBeachBanana]

Guys I'm trying to solve this problem but I'm having difficulties, here is what I tried:
$$\lim_{{x}\to 0} \frac{\sqrt[n]{c+x} - \sqrt[n]{c-x}}{x}, c>0, n\in \mathbb{N}$$

$$\lim_{{x}\to 0} \frac{\sqrt[n]{c+x} - \sqrt[n]{c-x}}{x} \times \frac{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}}{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}} = \lim_{{x}\to0}\frac{2}{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}}$$
how should I continue or should I use another approach?

Try L'Hopital

 

[topsquark]

Guys I'm trying to solve this problem but I'm having difficulties, here is what I tried:
$$\lim_{{x}\to 0} \frac{\sqrt[n]{c+x} - \sqrt[n]{c-x}}{x}, c>0, n\in \mathbb{N}$$

$$\lim_{{x}\to 0} \frac{\sqrt[n]{c+x} - \sqrt[n]{c-x}}{x} \times \frac{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}}{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}} = \lim_{{x}\to0}\frac{2}{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}}$$
how should I continue or should I use another approach?

...or the binomial expansion. (Look under "Generalizations.")

-Dan

 

[Albi]

Try L'Hopital

I'm not allowed to use that yet

 

[Dr.Peterson]

Guys I'm trying to solve this problem but I'm having difficulties, here is what I tried:
$$\lim_{{x}\to 0} \frac{\sqrt[n]{c+x} - \sqrt[n]{c-x}}{x}, c>0, n\in \mathbb{N}$$

$$\lim_{{x}\to 0} \frac{\sqrt[n]{c+x} - \sqrt[n]{c-x}}{x} \times \frac{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}}{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}} = \lim_{{x}\to0}\frac{2}{\sqrt[n]{c+x}^{n-1}+ \cdots + \sqrt[n]{c-x}^{n-1}}$$
how should I continue or should I use another approach?

What you showed is correct, though it would be clearer what you did if you had shown more terms, and more details of what you did.

To continue, you can just substitute 0 for x, since the expression is continuous.

If you really want another method, and you have learned about derivatives (in particular, the symmetric derivative), you could recognize that this limit is essentially that. That's how I checked your answer.

 

[blamocur]

To expand on @Dr.Peterson's post: the limit of the denominator in your last expression can be evaluated in a straightforward way.