# Evaluating undefined limits

1. Evaluate

2. lim χ→0 [((χ+1)^1/3) -1] / χ

3. The hint is [a^2 + b^2 = (a+b)(a^2 + ab + b^2)].
But i don't get it how can that help me solve this because it is a cube root and not to the power of 3.

Without thinking about the hint, I have attempted it and i think it is undefined for the function.
Say f(x) is the function, so f(0) is undefined making the limit as x approaches 0 undefined as well.
is it correct??

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Ray Vickson
Homework Helper
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1. Evaluate

2. lim χ→0 [((χ+1)^1/3) -1] / χ

3. The hint is [a^2 + b^2 = (a+b)(a^2 + ab + b^2)].
But i don't get it how can that help me solve this because it is a cube root and not to the power of 3.

Without thinking about the hint, I have attempted it and i think it is undefined for the function.
Say f(x) is the function, so f(0) is undefined making the limit as x approaches 0 undefined as well.
is it correct??

The hint is incorrect: a^2 + b^2 has 'a' and 'b' to the second power, while the right-hand-side has them to the third power. A correct hint would be a^3 - b^3 = (a-b)(a^2 + ab + b^2).

Think of setting y = (x+1)^(1/3), so the numerator of your expression is y-1. Can you see how to relate y-1 to y^3 - 1? Can you see how to write the denominator (x) in terms of y? Can you see what happens to y when x --> 0?

RGV

Think of setting y = (x+1)^(1/3), so the numerator of your expression is y-1. Can you see how to relate y-1 to y^3 - 1? Can you see how to write the denominator (x) in terms of y? Can you see what happens to y when x --> 0?

That's one possibility and it works fine.
But I prefer to multiply numerator and denominator with $a^2+ab+b^2$ for suitable a and b. What do you get if you use the (correct version of the) hint?

sorry I was wrong, the hint is: a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Ok, I did L'hopitals rule to derive the formula, until I am able to get a value from substituting zero in place of x. My answer came to 1/3. But still the hint had nothing to do with it.

Are you saying for me to do the opposite operation??

SammyS
Staff Emeritus
Homework Helper
Gold Member
sorry I was wrong, the hint is: a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Ok, I did L'hopitals rule to derive the formula, until I am able to get a value from substituting zero in place of x. My answer came to 1/3. But still the hint had nothing to do with it.

Are you saying for me to do the opposite operation??
Here is how the hint might be used:

The numerator $\sqrt{x+1}-1$. If you could get an equivalent expression with a numerator of $\displaystyle \left(\sqrt{x+1}\right)^3-1^3\,,$ that would allow major simplification including cancelling a factor that goes to zero in the numerator & denominator.

Use the hint to find out what you need to multiply $\sqrt{x+1}-1$ by in order to get $\displaystyle \left(\sqrt{x+1}\right)^3-1^3\ .$

By the way, welcome to PF!

so then I would have to multiply it with ($\sqrt{x+1}$)^2 + $\sqrt{x+1}$ +1
and then do the same with the numerator??
is it??

and afterwards derive it???

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SammyS
Staff Emeritus
Homework Helper
Gold Member
so then I would have to multiply it with ($\sqrt{x+1}$)^2 + $\sqrt{x+1}$ +1
and then do the same with the numerator??
is it??

and afterwards derive it???
No, you should not need to use L'Hôpital's rule.

Just multiply the numerator & denominator by $\displaystyle \left(\sqrt{x+1\ }\right)^2+\sqrt{x+1\ }+1\ .$ Then simplify.

oh YES

solved!

other than simplifying we can just substitute 0, right?