Question

Evaluate the limit, if it exists. (if an answer does not exist, enter dne.)lim h → 0 (x + h)3 − x3h

Answer 1

Presumably, the limit is

$\displaystyle\lim_{h\to0}\frac{(x+h)^3-x^3}h$

Now, if you're familiar with the definition of the derivatives, you'll notice that this is the limit form of the derivative of the function $f(x)=x^3$, which you may also know to be $3x^2$. But let's assume you don't know that just yet, and that it's actually the result you intend to find.

Expand the numerator:

$\dfrac{(x+h)^3-x^3}h=\dfrac{(x^3+3x^2h+3xh^2+h^3)-x^3}h=\dfrac{3x^2h+3xh+h^3}h$

Now, when $h\neq0$, we can divide through by the lowest power of $h$. We can do this because we're considering the limit as $h$ is *approaching* 0, and not when it actually takes on the value of $h=0$.

$\dfrac{(x+h)^3-x^3}h=3x^2+3xh+h^2$

Now, as $h\to0$, we can see only the leading term remains, so that

$\displaystyle\lim_{h\to0}\frac{(x+h)^3-x^3}h=3x^2$

as expected.