Question

Please Help Quickly!!!

Find the limit if f(x) = x^3

Answer 1

Option b. 12

Explanation:

This exercise asks us to find the derivative of a function using the definition of a derivative.

Our function is
`f(x) = x^(3)`. Therefore:

`f(2+h) = (2+h)^(3)`

`f(2) = (2)^(3) = 8`

Then:

`\lim_(h \to \0) (f(2+h)-f(2))/(h)=\lim_(h \to \0) ((2+h)^(3)-8)/(h)`

Expanding:

`\lim_(h \to \0) ((2+h)^(3)-8)/(h) =\lim_(h \to \0) (8+ h^(3) +6h(2+h) -8)/(h) =\lim_(h \to \0) (h^(3) +6h(2+h))/(h)`

`\lim_(h \to \0) (h^(3)+ 6h(2+h))/(h) =\lim_(h \to \0) h^(2) + 6(2+h)`

Now, if x=0:

`\lim_(h \to \0) (f(2+h)-f(2))/(h) = (0)^(2) +6(2+0) = 12`