This limit represents the derivative of some function f at some number a. Select an appropriate f and a.
lim_(h->0) (root4(256+h)-4)/h
1
f(x) = x^4, a = 4
f(x) = x^1/4, a = 4
f(x) = x^1/4, a = 256
f(x) = √x, a = 256
f(x) = √x, a = 16
To find an appropriate f(x) and a value for the given limit, we need to determine the function and the number such that when we take the derivative of the function at the number, it will yield the given limit.
Given the limit: lim(h->0) ((√(256+h))-4)/h
Let's start by simplifying the expression:
= lim(h->0) ((√(256+h)) - 4)/h
We notice that the expression (√(256+h)) - 4 is similar to the difference quotient, which is used to define the derivative. So, we look for a function whose derivative will give us this difference quotient.
Let's consider f(x) = √x. If we find the derivative of f(x) using the power rule, we get f'(x) = (1/2) * (x^(-1/2)).
Now, let's find the derivative of f(x) and evaluate it at a = 256 to see if it matches the given limit:
f'(256) = (1/2) * (256^(-1/2))
= (1/2) * (1/√256)
= (1/2) * (1/16)
= 1/32
The derivative of f(x) = √x at x = 256 is 1/32, which is not equal to 1. So, f(x) = √x with a = 256 is not an appropriate choice.
Let's try another option. Consider f(x) = x^(1/4). If we find the derivative of f(x) using the power rule, we get f'(x) = (1/4) * (x^(-3/4)).
Now, let's find the derivative of f(x) and evaluate it at a = 4 to see if it matches the given limit:
f'(4) = (1/4) * (4^(-3/4))
= (1/4) * (1/∛4)
= (1/4) * (1/2)
= 1/8
The derivative of f(x) = x^(1/4) at x = 4 is 1/8, which matches the given limit of 1. Therefore, f(x) = x^(1/4) with a = 4 is an appropriate choice in this case.