Finding Limits of F(x) as x Approaches 64 and 0

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In summary, the purpose of finding the limit of a function as x approaches a specific value is to understand the behavior of the function at that particular value. The limit is calculated by evaluating the function at progressively closer values of x and analyzing the resulting values. There is a difference between one-sided and two-sided limits, with the former only considering one direction of approach and the latter considering both. The limit can be used to determine if a function is continuous at a specific value, with the limit existing and being equal to the function's value indicating continuity. Algebraic methods can be used in some cases to calculate the limit, but more complex situations may require numerical or graphical methods.
  • #1
Redoctober
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I spent a lot thinking of this but can't fingure out how to find the limit as it approach x=64
Plz need help :)

F(x) = (sqrt(x)-8)/((x)^(1/3)-4)

find f(x) lim x->64

also this question -

lim x-->0 ((x+1)^(1/3)-1)/((x+1)^(1/4)-1)

Thanks in advance
 
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  • #2
Change variables to the lowest common denominator and factor and cancel
(sqrt(x)-8)/((x)^(1/3)-4) ->(y3-23)/(y2-22)
where y=x^(1/6)
((x+1)^(1/3)-1)/((x+1)^(1/4)-1)=(y4-14)/(y3-13)
where y=(x+1)^(1/12)
 
  • #3
Thanks :D ! I though bout this way but i didnt know wat to put y as
 

1. What is the purpose of finding the limit of a function as x approaches a specific value?

The limit of a function as x approaches a specific value is used to determine the behavior of the function at that particular value. It helps us understand if the function has a finite value or if it approaches infinity as x gets closer to the specified value.

2. How is the limit of a function as x approaches a value calculated?

The limit of a function as x approaches a value is calculated by evaluating the function at values of x that are progressively closer to the specified value. The resulting values are then analyzed to determine the limit.

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

The one-sided limit only considers the behavior of the function as x approaches the specified value from one direction (either from the left or from the right). The two-sided limit, on the other hand, considers the behavior of the function as x approaches the specified value from both the left and right sides.

4. How can we use the limit of a function to determine if it is continuous at a specific value?

If the limit of a function at a particular value exists and is equal to the value of the function at that point, then the function is continuous at that value. However, if the limit does not exist or is not equal to the value of the function, then the function is not continuous at that value.

5. Can the limit of a function as x approaches a value be calculated using algebraic methods?

Yes, in some cases, the limit of a function as x approaches a value can be calculated using algebraic methods such as factoring, simplifying, or applying properties of limits. However, in more complex cases, numerical or graphical methods may be needed to calculate the limit accurately.

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