Question

Using the definition of derivative state the function, f(x) and the value of a for lim as h->0 of (square root of (121+h) -11)/h

Answer 1

if
f(x) = x^.5
then
f(x+h) = (x + h)^.5


[f(x+h) - f(x)]/h = [(x+h)^.5-x^.5]/h

binomial expansion of
(x+h)^.5 = x^.5 + .5 x^-.5 h ..... higher powers of h
so
[ x^.5 + .5 x^-.5 h ...-x^.5 ]/h

or
.5 x^-.5 + higher powers of h

= .5 x^-.5 as h --->0

if x = 121
f(x) = 121^.5 = 11
f(x+h) = (121+h)^.5

df(121)/dx = .5 (121)^-.5 = .5/11

f(x) = 121