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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):
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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