Question 969699
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:


{{{graph(600,600,-5,5,-2,2,(sqrt(4x+1)-3)/(x-2),2/3)}}}


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.