`lim_(x->4) (x-4)/(x^3-64)` show steps to derive the limit of `1/48`

### 1 Answer | Add Yours

You should evaluate the limit converting the difference of cubes `x^3 - 64` into a product, using the following formula, such that:

`a^3 - b^3 = (a - b)(a^2 + ab + b^2)`

Reasoning by analogy yields:

`x^3 - 64 = (x - 4)(x^2 + 4x + 16)`

Replacing `(x - 4)(x^2 + 4x + 16)` for `x^3 - 64` yields:

`lim_(x->4) (x - 4)/((x - 4)(x^2 + 4x + 16))`

Reducing duplicate factors yields:

`lim_(x->4) 1/(x^2 + 4x + 16)`

You may evaluate a limit performing the following steps, such that:

`lim_(x->a) f(x) = f(a)`

Reasoning by analogy yields:

`lim_(x->4) 1/(x^2 + 4x + 16) = 1/(4^2 + 4*4 + 16)`

`lim_(x->4) 1/(x^2 + 4x + 16) = 1/48`

**Hence, evaluating the given limit yields**` lim_(x->4) 1/(x^2 + 4x + 16) = 1/48.`

### Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes