Evaluating a Limit: Examining lim h->0 ((8+h)^⅓ -2)/h

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Homework Help Overview

The discussion revolves around evaluating the limit as h approaches 0 for the expression ((8+h)^(1/3) - 2)/h. The subject area is calculus, specifically focusing on limits and derivatives.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the substitution of 8+h with x^3 and question the implications of this substitution as h approaches 0. There are inquiries about the reasoning behind certain transformations and the relationship between x and h.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the limit and its connection to derivatives. Some guidance has been offered regarding the nature of the limit, but no consensus or final resolution has been reached.

Contextual Notes

There are indications of uncertainty regarding the expected methods for solving the limit, and participants express concerns about the completeness of their calculations. Additionally, there are reminders about posting formats to ensure clarity in communication.

Aviegaille
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Homework Statement


Evaluate lim h->0 ((8+h)^⅓ -2)/h.

Homework Equations


Hint: Let 8+h=x^3

The Attempt at a Solution


I've uploaded a picture of my calculation. But I am not sure if that is the final answer or is there a following step to get the answer.
 

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Why do you think that as h->0 with x^3=8+h that x->(-3)? And why do you think that (x^3)^(1/3)=x^3?
 
Aviegaille said:

Homework Statement


Evaluate lim h->0 ((8+h)^⅓ -2)/h.

Homework Equations


Hint: Let 8+h=x^3

The Attempt at a Solution


I've uploaded a picture of my calculation. But I am not sure if that is the final answer or is there a following step to get the answer.

Please do not post thumbnails; they cannot be viewed on some media! Just type out things directly.
 
I don't know exactly which of many possible methods you are expected to use here, but did you notice that this limit is precisely that defining the derivative of f(x) at x= 8, with [itex]f(x)= x^{2/3}[/itex]?
 

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