In the limit #lim sqrtx=0# as #x->0^+#, how do you find #delta>0# such that whenever #0<x<delta#, #sqrtx<0.01#?

As #y=sqrtx# is a strictly increasing function:

#0 < x < epsilon^2 <=> 0 < sqrt(x) < sqrt(epsilon^2)#

So, for any #epsilon > 0#, if we take #delta_epsilon = epsilon^2# we have:

#0 < x < delta_epsilon <=> 0 < sqrt(x) < epsilon#

In particular for #epsilon =0.01# we can choose #delta_epsilon <= 0.0001#