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

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That might suggest using the rule for the derivative of a power function. Alternatively, you could use the binomial theorem to find an exact value for f(8+h)- f(8) for any h, and then divide by h. Either way, you should get an answer of 1/3.
  • #1
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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  • #2
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?
 
  • #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.
 
  • #4
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]?
 

1. What is the definition of a limit?

The limit of a function f(x) as x approaches a is the value that f(x) approaches as x gets closer and closer to a, but may not necessarily be equal to at a.

2. How do you evaluate a limit using algebraic manipulation?

To evaluate a limit, you can use algebraic manipulation to simplify the expression until you can directly substitute the value of the limit into the simplified expression. In this case, you can use properties of exponents to simplify the numerator, and then factor out the common h in the denominator.

3. What is the purpose of evaluating a limit?

Evaluating a limit allows us to find the behaviors of a function as x approaches a certain value. It can help us determine if a function is continuous or discontinuous at a specific point, and is an important concept in calculus and other areas of math and science.

4. Can a limit not exist?

Yes, a limit may not exist if the function has a vertical asymptote or if the left and right-hand limits approach different values. In this case, the limit is said to be undefined.

5. How do you evaluate a limit using the graph of a function?

You can evaluate a limit using the graph of a function by finding the y-value of the point on the graph that corresponds to the limit's value. This is the value that the function approaches as x approaches the given value. Alternatively, you can also use the graph to estimate the limit by looking at the behavior of the function near the given value.

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