limit x->1 (x^3-1)/(x-1) — step-by-step solution

Compute the limit of

\frac{x^{3} - 1}{x - 1}

x \to 1

The limit of

x^{3}

1

when

x \to 1

The limit of

x^{3} - 1

0

when

x \to 1

The limit of

x - 1

0

when

x \to 1

For

\frac{x^{3} - 1}{x - 1}

, the limit takes the indeterminate form 0/0 when

x

→

1

Let's apply L'Hôpital's rule (limit of a/b = limit of a'/b') by differentiating numerator and denominator:

Let's perform the differentiation:

{(x^{3} - 1)}^{'}

{(x^{3})}^{'} + 0

3 x^{2}

Let's perform the differentiation:

{(x - 1)}^{'}

1 + 0

1

With the derivatives of the numerator and the denominator, we obtain a new fraction whose limit can be determined:

The limit of

x^{2}

1

when

x \to 1

The limit of

3 x^{2}

3

when

x \to 1

So the limit of

\frac{x^{3} - 1}{x - 1}

3

when

x \to 1

▷Answer: $3$