Finding limits and using ε-δ definition.?

Question

How do I find the limit, L, for:

lim√x
x => 16

then how do i use the ε-δ definition to prove that L is the limit?

Answer 1

Well you _find_ it the same way you did in calculus, by simply substituting 16 into the equation to get [x→16]lim √x = √16 = 4. To prove that it is the limit, generally you pick ε>0, and then try to work out how close x has to be to 16 in order for √x to be within ε of 4. So we might start with:

|√x - 4|<ε
|x-16|/|√x+4|<ε
|x-16| < ε (√x+4)

Now, it's relatively easy to see that √x+4 is aways greater than, say, 1, so if |x-16|<ε, then certainly |x-16|< ε(√x+4). So now all that's left is to write the proof:

Let ε>0. Choose δ=ε, then 0<|x-16|<δ ⇒ |x-16|<ε ⇒ |x-16|<ε(√x+4) ⇒ |(x-16)/(√x+4)|<ε ⇒ |√x - 4| < ε, thus ∀ε>0, ∃δ>0 (specifically, δ=ε) such that 0<|x-16|<δ ⇒ |√x - 4|<ε, so [x→16]lim √x = 4. Q.E.D.

Edit: corrected typo.