limit as x approaches 3 of (x-1)/(x^2-1)

1) Compute the limit of

\frac{x - 1}{x^{2} - 1}

x \to 3

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

x - 1

3 - 1

when

x \to 3

⇥1.1) Let's evaluate:

3 - 1

⇥ⓘ Start from

3

and move back

1

on the number line:

3 - 1

2

⇥

2

ⓘ The power with a constant exponent is continuous here: substitute the limit of the base

The limit of

x^{2}

3^{2}

when

x \to 3

⇥1.2) Let's calculate:

3^{2}

⇥ⓘ An integer power is a repeated product:

3^{2}

3 \cdot 3

9

⇥

9

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

x^{2} - 1

9 - 1

when

x \to 3

⇥1.3) Let's evaluate:

9 - 1

⇥ⓘ Start from

9

and move back

1

on the number line:

9 - 1

8

⇥

8

So the limit of

\frac{x - 1}{x^{2} - 1}

\frac{2}{8}

when

x \to 3

⇥1.4) Let's evaluate:

\frac{2}{8}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{2}{8}

\frac{1 \cdot 2}{4 \cdot 2}

\frac{1}{4}

▶Limit:

\frac{1}{4}

2) The numeric value of:

\frac{1}{4}

is:

▷

0.25