# How to evaluate a limit with roots

## Homework Statement

I have the limit: lim [ (x+h)^2/3 - x^(2/3) /h ]

How would I further simplify and evaluate this limit.

2. The attempt at a solution

I have tried using a change of variable and using this in the sum of cubes formula (i.e. (x+h)^(2/3) = a, x^(2/3) = b, and then plugging this in to the sum of a cube equation. This did not help and just further complicated matters.

Next, I tried multiplying the numerator and denominator by {x+h)^(1/3) + x^(1/3) to cancel out the roots, but this just made new ones that were even harder to factor out.

I keep trying variations of the two methods I just described, but none of them work. Any direction or help on what to do next would be great. Thanks!

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Dick
Homework Helper

## Homework Statement

I have the limit: lim [ (x+h)^2/3 - x^(2/3) /h ]

How would I further simplify and evaluate this limit.

2. The attempt at a solution

I have tried using a change of variable and using this in the sum of cubes formula (i.e. (x+h)^(2/3) = a, x^(2/3) = b, and then plugging this in to the sum of a cube equation. This did not help and just further complicated matters.

Next, I tried multiplying the numerator and denominator by {x+h)^(1/3) + x^(1/3) to cancel out the roots, but this just made new ones that were even harder to factor out.

I keep trying variations of the two methods I just described, but none of them work. Any direction or help on what to do next would be great. Thanks!
Your sum of cubes formula really should help. (a-b)(a^2+ab+b^2)=a^3-b^3. Show why it doesn't? Multiply numerator and denominator by (a^2+ab+b^2).

Your sum of cubes formula really should help. (a-b)(a^2+ab+b^2)=a^3-b^3. Show why it doesn't? Multiply numerator and denominator by (a^2+ab+b^2).
Well I had h in the denominator, and what I was doing never allowed me to factor out the h in the expression. I ended up getting limh→0 [(a^(1/3) - b^(1/3))(a^2/3) + a^(1/3)b^(1/3) + b^(2/3) / h] and a = (x+h)^(2/3) and b = x^(2/3)

It got very messy, and I'm not sure if it's because I made an error in my steps, but I couldn't really go much further than this. I could just never factor out the h when I substituted the values for a and b back in. Any advice/thoughts would be appreciated.

Simon Bridge
Homework Helper
$$\lim_{h\rightarrow 0}\frac{(x+h)^{2/3}-x^{2/3}}{h}$$ ... is the derivative of ##x^{2/3}##.

Is the problem to show this derivative by the definition or just to show the derivative?

if the latter then you can rewrite it as ##y^3(x)=x^2## and use the power rule.

If the former - OK: then you need to check your algebra if you cannot get h to factor out.
Just looking at it - for some reason you ended up with h alone in the denominator ... don't do that. Leave the denominator alone and concentrate on factoring an h out of the numerator.

Compare yours with:

$$\lim_{h\rightarrow 0}\frac{(x+h)^{2/3}-x^{2/3}}{h}= \lim_{h\rightarrow 0}\left [ \left ( \frac{(x+h)^{2/3}-x^{2/3}}{h}\right ) \left ( \frac{(x+h)^{4/3}+(x+h)^{2/3}x^{2/3}+x^{4/3}}{(x+h)^{4/3}+(x+h)^{2/3}x^{2/3}+x^{4/3}}\right ) \right ]$$

... it does get messy - which is why we don't normally do it that way - you just have to slog through carefully. I find it helps to use a large window and a whiteboard marker.

1 person
Dick
Homework Helper
Compare yours with:

$$\lim_{h\rightarrow 0}\frac{(x+h)^{2/3}-x^{2/3}}{h}= \lim_{h\rightarrow 0}\left [ \left ( \frac{(x+h)^{2/3}-x^{2/3}}{h}\right ) \left ( \frac{(x+h)^{4/3}+(x+h)^{2/3}x^{2/3}+x^{4/3}}{(x+h)^{4/3}+(x+h)^{2/3}x^{2/3}+x^{4/3}}\right ) \right ]$$

... it does get messy - which is why we don't normally do it that way - you just have to slog through carefully. I find it helps to use a large window and a whiteboard marker.
It's not even all that much of a slog, is it? If you remember (a-b)(a^2+ab+b^2)=a^3-b^3, you can immediately replace the numerator with a^3-b^3=(x+h)^2-x^2. That's easy to expand. Then factor out the h and take h->0.

Last edited:
Simon Bridge