find the limit as x approaches zero for (2+x)^3 -8/x

undefined due to 0 in denominator

Actually, we have 0/0

lim[(2+x)^3 - 8]/x
= lim(3(2+x)^2)/1
= 3(4) = 12

Or, you can expand to get

(8 + 12x + 12x^2 + x^3 - 8)/x
= (12x + 12x^2 + x^3)/x
= 12 + 12x + x^2
= 12 as x->0

To find the limit as x approaches zero for the expression (2+x)^3 - 8/x, we can use algebraic manipulations and the concept of limits.

First, let's simplify the expression by expanding (2+x)^3 using the binomial theorem:
(2+x)^3 = 2^3 + 3(2^2)x + 3(2)(x^2) + x^3 = 8 + 12x + 6x^2 + x^3

Next, let's rewrite the expression by combining like terms:
(8 + 12x + 6x^2 + x^3) - (8/x)

Now, we can simplify further by canceling out the common factor of 8:
12x + 6x^2 + x^3 - (8/x)

In order to evaluate the limit as x approaches zero, we substitute x = 0 into the simplified expression:
lim(x->0) (12x + 6x^2 + x^3 - (8/x))

When we plug in x = 0, we get:
lim(x->0) (0 + 0 + 0 - (8/0))

At this point, we encounter a problem. Division by zero is undefined, and we cannot directly evaluate the limit. However, we can use some additional techniques to find the answer.

Observing the expression, we can see that (8/0) appears in the denominator. In general, whenever we have a rational expression with a zero in the denominator, it suggests a possible vertical asymptote or removable discontinuity.

Considering the term (8/x), as x approaches zero, the denominator becomes smaller and smaller (approaching zero), while the numerator remains constant (8). This indicates that the expression approaches positive infinity (∞) or negative infinity (-∞).

In conclusion, the limit as x approaches zero for the given expression is undefined or does not exist due to a vertical asymptote or removable discontinuity.