limit/derivative problem: limit [x->a] of (x^(1/3)-a^(1/3))/(x-a)

[ChuckTingle]

Evaluate

lim x^1/3 - a^1/3
x->a x-a

or

lim
x approaches a : (x^1/3-a^1/3)/(x-a)

Tried to do a change of variable but went wrong somewhere along the line.

 

[Deleted member 4993]

Evaluate

lim x^1/3 - a^1/3
x->a x-a

or

lim
x approaches a : (x^1/3-a^1/3)/(x-a)

Tried to do a change of variable but went wrong somewhere along the line.

Use:

c³ - d³ = (c - d)(c² + cd + d²)

 

[ChuckTingle]

Use:

c³ - d³ = (c - d)(c² + cd + d²)

This is what I got
let x^1/3=c and a^1/3=d

c-d
c³-d³
= c-d
(c - d)(c² + cd + d²)
= 1
(c² + cd + d²)
From there I get stuck. Do I return my original variables and set x - >a?

 

[ksdhart2]

From there I get stuck. Do I return my original variables and set x - >a?

That seems like a very good next step to me. What happens when you do that?

 

[ChuckTingle]

That seems like a very good next step to me. What happens when you do that?

I get
1
x^2/3+x^1/3a^1/3+a^2/3

then as x approaches a

1
a^2/3+a^1/3a^1/3+a^2/3

= 1
2a^2/3+a
This answer just seems weird to me, the evaluate questions where you don't find a value for a variable have always bothered me

 

[ksdhart2]

I get
1
x^2/3+x^1/3a^1/3+a^2/3

then as x approaches a

1
a^2/3+a^1/3a^1/3+a^2/3

= 1
2a^2/3+a
This answer just seems weird to me, the evaluate questions where you don't find a value for a variable have always bothered me

Your answer of 1/(2a^2/3+a) is almost right, but not quite. If you double check your math you'll find that a^1/3 * a^1/3 = (a^1/3)² is not a. As for your concerns purely symbolic answer, unfortunately, I don't know what to tell you. The correct answer to this problem will be in terms of a.

 

[ChuckTingle]

Your answer of 1/(2a^2/3+a) is almost right, but not quite. If you double check your math you'll find that a^1/3 * a^1/3 = (a^1/3)² is not a. As for your concerns purely symbolic answer, unfortunately, I don't know what to tell you. The correct answer to this problem will be in terms of a.

Thank you, I believe I get it now. It's been a while and I just need to get through the rust.