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I am stuck on evaluating the limit of a rational function

  • #1

Homework Statement



Evaluate the limit: [tex]lim_{(x)\rightarrow(2)}\frac{x^2+x-6}{sqrt(x+4)-sqrt(6)}[/tex]

Homework Equations



N/A

The Attempt at a Solution



I've drawn the graph which indicates that the at x=2, y=0, so 0 would seem to be the limit.
I could not, however, get the limit expression by algebraic manipulation to get to 0.

First of all, I rationalized the denominator to give x-2, which is a common factor of the numerator and the denominator. But canceling this, I am left with the expression:

[tex]lim_{(x)\rightarrow(2)}((x+3)(sqrt(x+4)+sqrt(6))[/tex]

after the [tex]lim_{(x)\rightarrow(2)}\frac{x-2}{x-2}[/tex] comes to 1.

I may have missed something crucial, so I just can't get the limit to be 0

Any help is greatly appreciated,
Charismaztex
 

Answers and Replies

  • #2
statdad
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Try again - the limit is not zero. Your simplification is correct.
 
  • #3
HallsofIvy
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That is NOT, by the way, a rational function.
 
  • #4
@statdad The graph shows that at x=2, y=0. How is this explained?

@ HallsofIvy Why not? I thought it can be expressed in the form p/q where p and q are real. Or if there are irrational numbers anywhere, does this make it void?
 
  • #5
Hurkyl
Staff Emeritus
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A rational function is a ratio of polynomials.
 
  • #6
statdad
Homework Helper
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@statdad The graph shows that at x=2, y=0. How is this explained?
Check your graph again. the one i've attached doesn't show that: in fact, there is no point on the graph at x = 2 because of the denominator.
@ HallsofIvy Why not? I thought it can be expressed in the form p/q where p and q are real. Or if there are irrational numbers anywhere, does this make it void?
A rational number is a ratio p/q where both numerator and denominator are integers. A rational expression is a fraction in which both numerator and denominator are polynomials.
 

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