limit x->3 (x^2-9)/(x-3) — step-by-step solution

1) Let's evaluate the limit of

\frac{x^{2} - 9}{x - 3}

when

x \to 3

ⓘ 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

⇥1.1) Let's evaluate:

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} - 9

9 - 9

when

x \to 3

⇥1.2) Let's calculate:

9 - 9

⇥ⓘ Start from

9

and move back

9

on the number line, down to zero:

9 - 9

0

⇥

0

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

The limit of

x - 3

3 - 3

when

x \to 3

⇥1.3) Let's calculate:

3 - 3

⇥ⓘ Start from

3

and move back

3

on the number line, down to zero:

3 - 3

0

⇥

0

For

\frac{x^{2} - 9}{x - 3}

, we encounter the indeterminate form 0/0 as

x

tends to

3

2) To find the limit, we factorize the numerator and the denominator:

x^{2} - 9

For

x^{2} - 9

, we recognize a difference of two squares:

x^{2} - 3^{2}

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = x

and

b = 3

Factors identified:

(x + 3) \cdot (x - 3)

3) Let's simplify:

\frac{(x + 3) \cdot (x - 3)}{x - 3}

ⓘ The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor

x - 3

\frac{(x + 3) \cdot (x - 3)}{x - 3}

x + 3

x + 3

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

The limit of

x + 3

3 + 3

when

x \to 3

Let's evaluate:

3 + 3

ⓘ Start from

3

and move forward

3

on the number line:

3 + 3

6

6

So the limit of

\frac{x^{2} - 9}{x - 3}

6

when

x \to 3

▶Limit:

6