How do you solve limX->1(2-SQRT(x+3))\(2x-2) algebraically?

Question

limX->1
(2 - (SQRT(x+3))) \ (2x-2)
Sorry about the symbols, but it is the limit as x approaches 1 for the formula 2 minus the square root of (x+3) divided by the quantity (2x-2) I know how to solve for this using the table on the calculator, but how do I solve in an algebraic manner, if you are smart and can help, I would appreciate it a whole bunch. Thanks

Answer 1

multiply both the numerator and denominator by 2+sqrt(x+3)

so we will have 4 - (x+3)/2(x-1)(2+sqrt(x+3))

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

-(x-1)/2(x-1)(2+sqrt(x+3)), (x-1) cancels

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

substituting

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

-1/2(2+2)

-1/8

Answer 2

1

Answer 3

This can be solved with calculus. A limit problem of the form 0 / 0 can be solved by taking the derivative of the numerator and dividing by the derivative of the denominator.
First the derivative of the numerator is
-1
-------------------- = -1/4 when x = 1
2 x sqrt(X + 3)

next the denominator

2 which is a constant for any value of x.

now

-1/4
---------- = -1/8
2

Answer 4

You could use l'Hopital's rule, which says that if the limit goes to 0/0, you can take the derivative of the top and bottom and get the same limit. In this case, that would leave you with:

[0- (1/2)(x+3)^(-1/2) ] / (2-0) =
-(x+3)^(-1/2) / 4 =
- 1 / 4(x+3)^(1/2)

The limit of this as x->1 is -1 / 4(4)^(1/2) = -1/8.

Answer 5

Well, try replacing sqrt(x + 3) with t.
x->1 becomes x+3->4 becomes sqrt(x+3) ->2.
We get the limit as t approaches 2 of (2-t)/(2 t^2 - 8)
This is negative one half of the limit (t -> 2) (t-2)/(t^2-4)
If you factor t^2-4, you get (t+2)(t-2). Now
(-1 / 2) limit (t -> 2) (t-2)/((t+2)(t-2))
Cancel the t-2, and you get 1/(t+2), which goes to one quarter when t goes to 2.
Your answer, then, is -1/8

Answer 6

lim x→1 of [2 - √(x + 3)]/(2x - 2)
= lim x→1 of [2 - √(x + 3)][2 + √(x + 3)]/{2(x - 1)[2 + √(x + 3)]}
= lim x→1 of (1 - x)/{2(x - 1)[2 + √(x + 3)]}
= lim x→1 of -1/{2[2 + √(x + 3)]}
= lim x→1 of -1/{2[2 + √(1 + 3)]}
= lim x→1 of -1/{2(2 + 2)} = -1/8