Evaluate the limit lim x->2 of (x^2 - 4)/(x - 2).

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Question: Evaluate the limit lim x->2 of (x^2 - 4)/(x - 2).

Options:

Correct Answer: 4

Solution:

Factoring gives (x-2)(x+2)/(x-2) = x + 2, thus limit is 4.

Evaluate the limit lim x->2 of (x^2 - 4)/(x - 2).

Evaluate the limit lim x->2 of (x^2 - 4)/(x - 2).

Correct Answer: 4

Step 1: Identify the limit we want to evaluate: lim x->2 of (x^2 - 4)/(x - 2).
Step 2: Notice that if we plug in x = 2 directly, we get (2^2 - 4)/(2 - 2) = 0/0, which is undefined.
Step 3: Factor the numerator x^2 - 4. It can be factored as (x - 2)(x + 2).
Step 4: Rewrite the limit using the factored form: lim x->2 of ((x - 2)(x + 2))/(x - 2).
Step 5: Cancel the (x - 2) terms in the numerator and denominator, as long as x is not equal to 2. We get lim x->2 of (x + 2).
Step 6: Now, we can safely substitute x = 2 into (x + 2): 2 + 2 = 4.
Step 7: Therefore, the limit is 4.

Limit Evaluation – Understanding how to evaluate limits, particularly when direct substitution results in an indeterminate form.
Factoring – Using algebraic manipulation, such as factoring, to simplify expressions before evaluating limits.
Indeterminate Forms – Recognizing and resolving indeterminate forms like 0/0 that can occur in limit problems.