Limit of (x^3-1)/(x^1/2-1) as x approaches 1

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In summary, the conversation discusses how to convert a given fraction into an equivalent form by multiplying both the numerator and denominator by the conjugate of the denominator. This is done in order to factor out the 0 in the denominator and simplify the expression. The conversation also mentions a different approach using a substitution to solve the limit.
  • #1
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Homework Statement


lim x->1 (x^3-1)/(x^1/2-1)
ans:6

Homework Equations



The Attempt at a Solution


(x^1/2-1)^-1
can be converted into (plugged into wolfram):
(-x^1/2-1)/(1-x)
i want to how this done if I'm factorize out the 0 in the denominator
 
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  • #2
If I'm getting it right you have

Numerator X³ -1
Denominator Sqrt(x) - 1

How can you factorize X³-1? If you know how the answer stares you right in the face.
 
  • #3
ok i may have expressed myself wrongly there.
i wanted to know is how (1 / sqrt(x) - 1) can be converted into (-sqrt(x) - 1) / (1 - x)
noticed it was just multiplying denominator and numerator by its conjugate.
so yeah, should've noticed the elephant in the room.
thanks for the help anyway.
 
  • #4
same way as 1/sqrt2 is sqrt2/2

you lose square root in the denominator
1*(sqrt(x) +1)/ (sqrt(x) -1)(sqrt(x) +1)
sqrt(x) +1 / (x-1) , multiply both sides of the division sign by -1 and you arrive at what you are looking for.
 
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  • #5
tg43fly said:

Homework Statement


lim x->1 (x^3-1)/(x^1/2-1)
ans:6

Homework Equations



The Attempt at a Solution


(x^1/2-1)^-1
can be converted into (plugged into wolfram):
(-x^1/2-1)/(1-x)
i want to how this done if I'm factorize out the 0 in the denominator

Why not let t = x^(1/2), and so have the limit of (t^6 - 1)/(t-1) as t → 1?
 

What is a limit probability?

A limit probability is the probability of a certain event occurring as the number of trials approaches infinity. It is used to determine the long-term likelihood of an event happening.

How is limit probability calculated?

Limit probability is calculated using the formula lim n→∞ P(X≤n), where n is the number of trials and P(X≤n) is the cumulative probability of the event occurring in n trials.

What is the significance of limit probability in scientific research?

Limit probability is important in scientific research as it can help determine the probability of rare events or the likelihood of an event occurring in a large number of trials. It can also be used to understand the behavior of complex systems and predict future outcomes.

What are some examples of limit probability in real-life situations?

One example of limit probability is in genetics, where it can be used to predict the probability of a certain trait appearing in a large population. Another example is in weather forecasting, where limit probability can help predict the likelihood of extreme weather events occurring in a given area.

What are the limitations of limit probability?

Limit probability assumes that the trials are independent and identically distributed, which may not always be the case in real-life situations. It also does not take into account any external factors that may affect the outcome of an event. Additionally, limit probability is based on theoretical calculations and may not always accurately reflect real-world scenarios.

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