I am stuck on evaluating the limit of a rational function

[Charismaztex]

Homework Statement

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

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:

$$lim_{(x)\rightarrow(2)}((x+3)(sqrt(x+4)+sqrt(6))$$

after the $$lim_{(x)\rightarrow(2)}\frac{x-2}{x-2}$$ 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

 

[statdad]

Try again - the limit is not zero. Your simplification is correct.

 

[HallsofIvy]

That is NOT, by the way, a rational function.

 

[Charismaztex]

@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?

 

[Hurkyl]

A rational function is a ratio of polynomials.

 

[statdad]

@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.