Posted by DJ on Tuesday, October 4, 2011 at 10:06am.
show lim x>3 (sqrt(x)) = (sqrt(c))
hint: 0< sqrt(x)sqrt(c)= (xc)/(sqrt(x)+sqrt(c))<(1/(sqrt(c))*xc

Calculus  Count Iblis, Tuesday, October 4, 2011 at 2:29pm
lim x>c (sqrt(x)) = (sqrt(c))
means that the closer x is chosen to c the closer sqrt(x) comes to sqrt(c). To prove that, you need to show that making sqrt(x)  sqrt(c) smaller than some arbitrary epsilon is always possible by restricting x to some interval around c. Since epsilon can be any number larger than zero, this means that this interval around zero exists no matter how small epsilon is chosen. So, even if epsilon = 0.000000000001, there is stll a small interval around c for which the square root function is within 0.000000000001 of c.
From the inequalities:
0< sqrt(x)sqrt(c)= (xc)/(sqrt(x)+sqrt(c))<(1/(sqrt(c))*xc
you can see that to require
sqrt(x)sqrt(c) < epsilon
it is enough to require that:
xc < sqrt(c) epsilon
So, we see that for arbitrary
epsilon > 0 there indeed does exist a delta > 0 such that for xc < delta we have that
sqrt(x)  sqrt(c) < epsilon
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