What is the value of the limit lim (x→2) (x² - 4)/(x - 2)?

To find the limit ( \lim_{{x \to 2}} \frac{x^2 - 4}{x - 2} ), we first notice that directly substituting ( x = 2 ) into the expression results in the indeterminate form ( \frac{0}{0} ). This indicates the need for further manipulation.

The expression ( x^2 - 4 ) can be factored using the difference of squares. It factors as:

[

x^2 - 4 = (x - 2)(x + 2)

]

Substituting this factorization back into the limit gives us:

[

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

]

As long as ( x \neq 2 ), we can safely cancel ( x - 2 ) from the numerator and denominator, simplifying the limit to:

[

\lim_{{x \to 2}} (x + 2)

]

Now, we can directly substitute ( x = 2 ) into the simplified expression:

[

2 + 2 = 4

]

Thus