18) lim (x^3 - 1) / (x - 1)
x - 1
x^2 - 1

Q: 18) lim (x^3 - 1) / (x - 1) x -> 1 x^2 - 1

The expression is: \[ \lim_{{x \to 1}} \frac{x^3 - 1}{x - 1} \] To solve this, we can factor the numerator using the difference of cubes formula: \[ x^3 - 1 = (x - 1)(x^2 + x + 1) \] Substitute this back into the limit expression: \[ \lim_{{x \to 1}} \frac{(x - 1)(x^2 + x + 1)}{x - 1} \] Cancel out the $(x - 1)$ terms: \[ \lim_{{x \to 1}} (x^2 + x + 1) \] Now, substitute $x = 1$: \[ 1^2 + 1 + 1 = 3 \] So, the limit is $\boxed{3}$.

Question

18) lim (x^3 - 1) / (x - 1)
      x -> 1
      x^2 - 1

Accepted Answer

The expression is:

$$
\lim_{{x \to 1}} \frac{x^3 - 1}{x - 1}
$$

To solve this, we can factor the numerator using the difference of cubes formula:

$$
x^3 - 1 = (x - 1)(x^2 + x + 1)
$$

Substitute this back into the limit expression:

$$
\lim_{{x \to 1}} \frac{(x - 1)(x^2 + x + 1)}{x - 1}
$$

Cancel out the $(x - 1)$ terms:

$$
\lim_{{x \to 1}} (x^2 + x + 1)
$$

Now, substitute $x = 1$:

$$
1^2 + 1 + 1 = 3
$$

So, the limit is $\boxed{3}$.

18) lim (x^3 - 1) / (x - 1) x -> 1 x^2 - 1

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