Question

lim_(x->59)(sqrt(x+5)-8)/(x-59)??

Answer 1

note that when we substitute 59 to the expression in numerator and denominator,
(sqrt(x+5)-8)/(x-59)
(sqrt(59+5)-8)/(59-59)
(sqrt(64)-8)/0
8-8/0
0/0
thus we use L'hopital's Rule: we separately get the derivative of numerator and denominator. thus,
(num) = (sqrt(x+5)-8)
(num) = (x+5)^(1/2) - 8
d(num) = (1/2)*(x+5)^(-1/2)
d(num) = 1/(2*sqrt(x+5))

(deno) = x-59
d(deno) = 1

and we rewrite it again.
lim as x->59 of 1/(2*sqrt(x+5)) / 1
we can now substitute 59 to x:
1/(2*sqrt(59+5))
1/(2*sqrt(64))
1/(2*8)
1/16

hope this helps~ :)