Guess the value of the limit

Question

Guess the value of the limit (if it exists) by evaluating the function at the given numbers.

Guess the value of the limit

Answer 1

Is there a way to find the limit without guessing?

One way is to realize that the numerator is a difference of cubes, i.e. (a³ - b³) = (a - b)(a² + ab + b²).

The original numerator can be factored using the difference of cubes:

(4+h)³ - 4³ = [(4+h) - 4][(4+h)² + 4(4+h) + 16] = h[(4+h)² + 4(4+h)+16]

For the original limit as h->0, the h in the numerator cancels with the h in the denominator (as long as h gets close to zero, but not equal to zero).

Now take the limit as h->0 by substituting 0 for h.

Here is how you can use Desmos to perform the tedious calculations:

desmos.com/calculator/fksovtzbl3