SOLUTION: Lim Sqrt{4x+1}-3 / x-2 x--->2 How do i do this. I cant figure out how to get rid of the sqrt. I looked online and everyone says to rationalize the numerator but they

Question 969699: Lim Sqrt{4x+1}-3 / x-2
x--->2
How do i do this. I cant figure out how to get rid of the sqrt. I looked online and everyone says to rationalize the numerator but they skip steps so idk what I'm supposed to do. Can someone explain without skipping a step even if it seems easy to you.
Found 2 solutions by rothauserc, Theo:
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
Is the expression really,
(Lim Sqrt{4x+1}-3) / (x-2) as x--->2 ?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
find the limit of sqrt(4x+1)-3 / (x-2) as x approaches 2.

i believe that what they told you is a correct way to look at it.

if you were able to graph the function, you would have found that the answer should be 2/3 or thereabouts.

that's just by eyeballing.

algebraically, you would do the following:

start with:

(sqrt(4x+1)-3) / (x-2)

multiply numerator and denominator by (sqrt(4x+1)+3) to get:

((sqrt(4x+1)-3)*(sqrt(4x+1)+3)) / ((x-2)*(sqrt(4x+1)+3))

simplify to get:

((4x+1)-9) / ((x-2)*(sqrt(4x+1)+3))

simplify further to get:

(4x-8) / ((x-2)*(sqrt(4x+1)+3))

factor out a 4 in the numerator to get:

4*(x-2) / ((x-2)*(sqrt(4x+1)+3))

the (x-2) in the numerator and denominator cancel out and you are left wtih:

4 / (sqrt(4x+1)+3))

when x = 2, this becomes 4 / (sqrt(9)+3)) which becomes 4 / (3+3) which becomes 4/6 which becomes 2/3.

that's your solution.

the limit of (sqrt(x+1)-3) / (x-2) as x approaches 2 is equal to 2/3.

in the graph, you can see that there is a hole at x = 2, and if you gave values of x as 1.999999999 and 2.000000001, you would see that they hovered about 2/3.

the graph looks like this:

the horizontal line is at y = 2/3.

you can see that the graph of y = 2/3 intersects with the graph of y = (sqrt(4x+1)-3)/(x-2) at x = 2.

you will not, however, be able to find that value since the function is undefined at x = 2.

there is a hole there that you can't see.

the hole is because when you try to evaluate the function at x = 2, the answer is undefined.

it does, however, show you that, as you approach x = 2, the answer will approach x = 2/3.

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