De-Rationalizing a Surd Fraction: Step-by-Step Guide

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In summary: Then the limit becomes \frac{sin^{-1}x}{x}=\frac{u}{sin(u)}=\frac{1}{\frac{sin(u)}{u}}=\frac{1}{1}=1In summary, the limit of the given expression is equal to 1, without using L'hopital's rule or the definition. This can be achieved by using the "conjugate trick" to simplify the expression and make it possible to substitute x=2. For the second question, the limit can be found by either using the series expansion or making the substitution x=sin(u).
  • #1
lkh1986
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Homework Statement



Find the lim of [itex]\frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}[/itex]when x approaches 2, without using L'hopital rule or the definition.

Homework Equations


The Attempt at a Solution


I try to multiply both the numerator and the denominator to get [itex]\frac{(\sqrt{6-x}-2)(\sqrt{3-x}-1)}{2-x}[/itex], but when I substitute x = 2, the term becomes undefined.

Any suggestion? Thanks.

I am thinking of de'rationalize' the rational surd fraction, but nothing works.
 
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  • #2
1. First off:
There should be a "+1" rather than "-1" in the remaining factor after you myliplied both denominator and numerator with the 2conjugate".

2. secondly:
do the same "conjugate trick", with the surd expression that is contained in the numerator.
 
  • #3
Oops, sorry. Now I have

[itex]\frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}[/itex]

=[itex]\frac{(\sqrt{6-x}-2)(\sqrt{3-x}+1)}{(\sqrt{3-x}-1))(\sqrt{3-x}+1)}[/itex]

=[itex]\frac{(\sqrt{6-x}-2)(\sqrt{3-x}+1)}{2-x}[/itex]
 
  • #4
Correct.
Now do the same trick with the other surd expression.
 
  • #5
Yay, I get it. I multiply by another conjugate and eventually the (2-x) term will cancel out. Then I will get 2/4 which is equal to 1/2. Thanks so much for the help. :)

[itex]\frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}[/itex]

=[itex]\frac{(\sqrt{6-x}-2)(\sqrt{3-x}+1)}{(\sqrt{3-x}-1))(\sqrt{3-x}+1)}[/itex]

=[itex]\frac{(\sqrt{6-x}-2)(\sqrt{3-x}+1)}{2-x}[/itex]

=[itex]\frac{(\sqrt{6-x}-2)(\sqrt{3-x}+1)(\sqrt{6-x}+2)}{(2-x)(\sqrt{6-x}+2)}[/itex]

=[itex]\frac{(\sqrt{3-x}+1)(2-x)}{(2-x)(\sqrt{6-x}+2)}[/itex]

=[itex]\frac{\sqrt{3-x}+1}{\sqrt{6-x}+2}[/itex]

So if I replace x = 2, I will get 2/4 = 1/2. :)
 
  • #6
:smile:
 
  • #7
Another question:

Find the limit of [itex]\frac{sin^{-1}x}{x}[/itex] when x approaches 0.

The answer given is 1.

Seems like I can't express arcsin explicitly. Any clue on how to start? :)
 
  • #8
Can I use the series expansion?

[itex]\frac{sin^{-1}x}{x}[/itex]

=[itex]\frac{x+\frac{X^3}{6}+\frac{3x^5}{40}+...}{x}[/itex]

=1+[itex]\frac{x^2}{6}+\frac{3x^4}{40}+...[/itex]

Then I can get limit = 1 when I substitute x = 0.

Can we get the same answer without using the series expansion thingy?
 
  • #9
Or, you could make the substitution x=sin(u), which is valid in the neighbourhood of x=0.
 

FAQ: De-Rationalizing a Surd Fraction: Step-by-Step Guide

What is a surd fraction and why is it important to de-rationalize it?

A surd fraction is a fraction that contains a square root or other irrational number in either the numerator or denominator. De-rationalizing a surd fraction means to remove the irrational number and express the fraction in its simplest form. This is important because it allows for easier manipulation and calculation of the fraction.

How do I identify a surd fraction?

A surd fraction can be identified by looking for square roots or irrational numbers in the numerator or denominator. For example, 3/√2 is a surd fraction because it contains a square root in the denominator. It is important to note that not all fractions with square roots are surd fractions, only those in which the square root cannot be simplified to a whole number.

What is the step-by-step process for de-rationalizing a surd fraction?

The step-by-step process for de-rationalizing a surd fraction is as follows:

Step 1: Identify the surd fraction and determine which part (numerator or denominator) contains the square root or irrational number.

Step 2: Multiply both the numerator and denominator by the square root or irrational number in the opposite part. This will result in a fraction with a rational number in both the numerator and denominator.

Step 3: Simplify the resulting fraction to its simplest form, if possible. If not, leave it in its current form.

Can a surd fraction be simplified without de-rationalizing it?

Yes, a surd fraction can be simplified without de-rationalizing it if the square root can be simplified to a whole number. For example, √18/√2 can be simplified to √9/√1, which is equal to 3/1. However, if the square root cannot be simplified to a whole number, de-rationalizing is necessary.

How can I use de-rationalization in real-life applications?

De-rationalization is commonly used in fields such as engineering, physics, and finance to simplify equations and calculations involving surd fractions. It is also used in everyday situations where fractions with irrational numbers need to be simplified, such as splitting a bill between friends or calculating measurements for construction projects.

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