calculate the limit of x->2 (x^2-4)/(x-2)

1) Compute the limit of

\frac{x^{2} - 4}{x - 2}

x \to 2

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

The limit of

x^{2}

2^{2}

when

x \to 2

⇥1.1) Let's evaluate:

2^{2}

⇥ⓘ A power with a positive integer exponent is a repeated product:

2^{2}

2 \cdot 2

4

⇥

4

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

The limit of

x^{2} - 4

4 - 4

when

x \to 2

⇥1.2) Let's evaluate:

4 - 4

⇥ⓘ Start from

4

and move back

4

on the number line, down to zero:

4 - 4

0

⇥

0

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

The limit of

x - 2

2 - 2

when

x \to 2

⇥1.3) Let's calculate:

2 - 2

⇥ⓘ Start from

2

and move back

2

on the number line, down to zero:

2 - 2

0

⇥

0

For

\frac{x^{2} - 4}{x - 2}

, we encounter the indeterminate form 0/0 as

x

tends to

2

2) To determine the limit, factor the numerator and denominator:

x^{2} - 4

For

x^{2} - 4

, we recognize a difference of two squares:

x^{2} - 2^{2}

. Apply

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

, with

a = x

and

b = 2

Factors identified:

(x + 2) \cdot (x - 2)

3) Let's simplify:

\frac{(x + 2) \cdot (x - 2)}{x - 2}

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

x - 2

\frac{(x + 2) \cdot (x - 2)}{x - 2}

x + 2

x + 2

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

The limit of

x + 2

2 + 2

when

x \to 2

Let's evaluate:

2 + 2

ⓘ Start from

2

and move forward

2

on the number line:

2 + 2

4

4

So the limit of

\frac{x^{2} - 4}{x - 2}

4

when

x \to 2

▶Limit:

4