Find the limit:?

Question

Can anyone help me how to solve this problem step by step. Please click on the link to see the problem. Thanks...

http://img528.imageshack.us/img528/697/18717776gk2.png

Answer 1

lim_{∆x → 0} [ √({x + ∆x} + 2) − √(x + 2) ] ⁄ [∆x] =

lim_{∆x → 0} [ √(x + ∆x + 2) − √(x + 2) ] ⁄ [∆x] =

Multiply top and bottom by 1, the conjugate of the numerator
lim_{∆x → 0} [ √(x + ∆x + 2) − √(x + 2) ]⋅[ √(x + ∆x + 2) + √(x + 2) ] ⁄ [∆x]⋅[ √(x + ∆x + 2) + √(x + 2) ] =

lim_{∆x → 0} [ (x + ∆x + 2) − (x + 2) ] ⁄ [∆x]⋅[ √(x + ∆x + 2) + √(x + 2) ] =

lim_{∆x → 0} [ ∆x ] ⁄ [∆x]⋅[ √(x + ∆x + 2) + √(x + 2) ] =

lim_{∆x → 0} 1 ⁄ [ √(x + ∆x + 2) + √(x + 2) ] =

Evaluate ∆x = 0
1 ⁄ [ √(x + {0} + 2) + √(x + 2) ] =

1 ⁄ [ 2⋅√(x + 2) ]



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The expression:
lim_{∆x → 0} [ √({x + ∆x} + 2) − √(x + 2) ] ⁄ [∆x]
takes the form of the limit definition of a derivative, for the function f(x) = √(x + 2).

So, you could simply find the derivative of √(x + 2).

Even though I noticed this right off the bat, I respect the fact that you may not yet know derivative calculus.

Furthermore, derivation is based on the evaluation of limits. It wouldnt be sound logic to prove the limit through differentiation, when limits are the basis for differentiation.

This is a criticism of Steiners answer. He obviously takes the concepts for granted. We need to respect the tree of conceptual progression.

Answer 2

This just the derivative of f(x) = sqrt(x +2). So, the answer is f'(x) = 1/(2sqrt(x +2))

Answer 3

difficult to type here
but the conjugate...

(a+b) (a-b) = a^2 - b^2

Answer 4

Maybe you have no life, and ought to go click yourself.