Limit of Fraction: x→1 (sqrt(x)−x2)/(1−sqrt(x))

I understand that the final answer is -2/3.In summary, the conversation begins with someone asking for the value of a limit. They then ask if it is possible for the value to be -2 and apologize for not being able to solve it themselves. Another person suggests factoring the bottom of the fraction and using a substitution method. The original person thanks them and asks for help with factoring a term. The conversation ends with someone providing the final answer of -2/3.
  • #1
nejnadusho
31
0
Hi guys I have a question?


lim x->1 (sqrt(x)-x^2)/(1-sqrt(x))

What is the value of the limit of this function?

Is it possible to be (-2)?

Or I AM WRONG?

Thank you in advance.
 
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  • #2
… just divide …

nejnadusho said:
(sqrt(x)-x^2)/(1-sqrt(x))

Hi nejnadusho! :smile:

The bottom of this fraction actually factors the top, so you can just divide it out, if you want … :smile:
 
  • #3
??

I am sorry I cannot do it.

I don't know how to do it
 
Last edited:
  • #4
It might help to let [itex] \sqrt{x} = u[/itex] so that it becomes

[tex] \frac{u-u^4}{1-u} = \frac{ u(u^3-1)}{u-1}[/tex]

Do you know how to factorize the u^3 -1 term?
 
  • #5
Thank you nOW I got it.

No I don't know how to factorize this kind of terms
 
  • #6
[tex]a^3-b^3 = (a-b)(a^2+ab+b^2)[/tex]
Can you proceed?
 
  • #7
Yes thanks
 

1. What does the limit of this fraction represent?

The limit of this fraction represents the value that the fraction approaches as x gets closer and closer to 1.

2. How do you solve for the limit of this fraction?

To solve for the limit, you can use algebraic manipulation and the concept of factoring to simplify the fraction and then plug in the value of 1 for x to find the limit.

3. Can this limit be evaluated by direct substitution?

No, because direct substitution would result in an indeterminate form of 0/0 which is undefined. This is why algebraic manipulation is necessary to solve for the limit.

4. What is the significance of the limit of this fraction?

The limit of this fraction is significant because it helps us understand the behavior of the function as x approaches 1. It can also help us determine if the function is continuous at x=1.

5. Can this limit have a different value for different values of x?

No, the limit of this fraction only has one value as x approaches 1. This is because the value of the limit is determined by the behavior of the function near x=1, not at x=1.

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