Step 1: Identify the limit we want to calculate: lim (x -> 2) (x^3 - 8)/(x - 2).
Step 2: Notice that when we plug in x = 2, both the numerator and denominator become 0. This means we have a 0/0 indeterminate form.
Step 3: To resolve this, we can factor the numerator x^3 - 8. This is a difference of cubes, which can be factored as (x - 2)(x^2 + 2x + 4).
Step 4: Rewrite the limit using the factored form: lim (x -> 2) ((x - 2)(x^2 + 2x + 4))/(x - 2).
Step 5: We can cancel the (x - 2) in the numerator and denominator, as long as x is not equal to 2. This simplifies our limit to: lim (x -> 2) (x^2 + 2x + 4).
Step 6: Now, we can directly substitute x = 2 into the simplified expression: 2^2 + 2*2 + 4.
Step 7: Calculate the result: 4 + 4 + 4 = 12.
Step 8: Therefore, the limit is 12.
Limit Calculation – Understanding how to evaluate limits, particularly when dealing with indeterminate forms.
Factoring Polynomials – The ability to factor polynomials to simplify expressions before taking limits.
L'Hôpital's Rule – Recognizing when to apply L'Hôpital's Rule for limits that result in indeterminate forms.
