What is the limit of $(x^2-4)/(x-2)$ as $x$ approaches 2?

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Answer 1 · 2014-10-18T05:12:42

A direct substitution results in the indeterminate form $0/0$ . We should then try to simplify the function.

In this example we see that the numerator is a difference of perfect squares.

Remember factoring rules back to Algebra I.

$(a^2-b^2)=(a-b)(a+b)$

In this example

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

$lim_(x->2) (x^2-4)/(x-2)=lim_(x->2) ((x-2)(x+2))/(x-2)$

Cancel out the factor $(x-2)$

$lim_(x->2) (x+2)$

Direct substitution

$(2+2) = 4$

What is the limit of (x^2-4)/(x-2)(x^2-4)/(x-2) as xx approaches 2?