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
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
To find the function and the value of "a" using the definition of derivative, we need to rewrite the given expression in a way that resembles the difference quotient form used in the definition.
The derivative of a function f(x) at a point "a" is defined as the limit of the difference quotient as "h" approaches 0:
f'(a) = lim(h->0) [f(a+h) - f(a)] / h
In this case, we are given the expression:
lim(h->0) [(√(121+h) - 11) / h]
To rewrite this expression in the difference quotient form, we can multiply and divide by the conjugate of the numerator, which will allow us to remove the square root:
lim(h->0) [(√(121+h) - 11) / h] * [(√(121+h) + 11) / (√(121+h) + 11)]
Expanding and simplifying, we get:
lim(h->0) [(121+h - 121) / (h(√(121+h) + 11))]
Simplifying further:
lim(h->0) [h / (h(√(121+h) + 11))]
Canceling out the "h" in the numerator and denominator:
lim(h->0) [1 / (√(121+h) + 11)]
Now, we can see that the function is f(x) = 1 / (√(121+x) + 11), and the value of "a" is 0.
So, f(x) = 1 / (√(121+x) + 11) and a = 0.