Find the limit of this function

In summary, the conversation is about finding the limit of a function using only algebraic manipulation without using Newton's Method or L'Hopital's Rule. The limit in question is for the function (^3√x - 4)/(^√x - 8) as x approaches 64. The solution involves factoring using the difference of two squares/cubes rule, although the use of x^(1/3) and x^(1/2) instead of x^2 and x^3 in the numerator and denominator may cause hesitation. The original poster also mentions previously posting the problem and receiving a solution using algebra, but not in the specific way they were looking for. Another poster comments on the impressive solution, noting that
  • #1
racer
39
0
Hello there

Find the lim of this function using Algebra alone and without using
Newton Method or L'Hapital Rule, only Algebraic manipulation is allowed.

I have previously posted this lim but no one did solve it algebriaclly, I solved it using
Algebra and uploaded the solution.

Lim [tex]\frac{^3\sqrt{x}- 4}{\sqrt{x}- 8}[/tex]
X ---> 64

The solution
http://img3.freeimagehosting.net/image.php?ae06b74212.jpg
 
Last edited by a moderator:
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  • #2
Thanks for the update, although I must have missed the original post. It is quite obvious to me that you would want to factor using the difference of two squares/cubes rule as you have 4 (a perfect square) and 8 (a perfect cube). The only thing that might make one hesitate is that instead of using x^2 and x^3 for the numerator and denominator respectively they use x^(1/3) and x(1/2).
 
  • #3
That's impressive! Of course, using L'Hopital's rule (which, I notice, you are not allowed to do) would make it fairly simple.
 
Last edited by a moderator:

What is the definition of a limit?

A limit is the value that a function approaches as its input variable gets closer and closer to a particular value.

How do I find the limit of a function algebraically?

To find the limit of a function algebraically, you can use techniques such as direct substitution, factoring, and rationalization. You can also use the limit laws, which state that the limit of a sum, difference, product, or quotient of two functions is equal to the sum, difference, product, or quotient of their limits, respectively.

What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the values of the function as the input variable approaches from one side (either the left or the right) of the particular value. A two-sided limit considers both sides and requires that the function approaches the same value from both directions.

Can I use a graph to find the limit of a function?

Yes, a graph can be a helpful tool in finding the limit of a function. You can visually observe the behavior of the function as the input variable approaches the particular value and make an estimate of the limit.

What are some common types of indeterminate forms when finding limits?

Some common types of indeterminate forms include 0/0, ∞/∞, 0∙∞, and ∞-∞. These forms require additional algebraic manipulation or the use of more advanced techniques such as L'Hôpital's rule to evaluate the limit.

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