Question

Find lim x →0 f(2+h)-f(2)/h if f(x)=x^3

Answer 1

Answer:

The value of the limit is 12.

Step-by-step explanation:

Small typing mistake, the limit is of h tending to 0.

We have that:

$f(x) = x^{3}$

Then

$f(2+h) = (2+h)^{3} = 8 + 12h + 6h^{2} + h^{3}$

$f(2) = 2^{3} = 8$

Calling the limit L

$L = \frac{f(2+h) - f(2)}{h}$

$L = \frac{8 + 12h + 6h^{2} + h^{3} - 8}{h}$

$L = \frac{h^{3} + 6h^{2} + 12h}{h}$

h is the common term in the numerator, then

$L = \frac{h(h^{2} + 6h + 12)}{h}$

Simplifying by h

$L = h^{2} + 6h + 12$

Since h tends to 0.

$L = 0^{2} + 6*0 + 12$

$L = 12$

So the answer is 12.