Question

Question #d90f5

Answer 1

`d) f(x)=x^3, c=3`

Explanation:

The definition of a derivative of a function `f(x)` at a point `c` can be written:
`lim_(h->0)(f(c+h)-f(c))/h`

In our case, we can see that we have `(3+h)^3`, so we might guess that the function is `x^3`, and that `c=3`. We can verify this hypothesis if we write `27` as `3^3`:
`lim_(h->0)((3+h)^3-27)/h=lim_(h->0)((3+h)^3-3^3)/h`

We see that if `c=3`, we would get:
`lim_(h->0)((c+h)^3-c^3)/h`

And we can see that the function is just a value cubed in both cases, so the function must be `f(x)=x^3`:
`lim_(h->0)((text(///))^3-(text(//))^3)/h`