For f(x)=(x^2-4)/(x-2), x != 2, what is lim x->2 f(x)?

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Multiple Choice

For f(x)=(x^2-4)/(x-2), x != 2, what is lim x->2 f(x)?

Explanation:
When a limit is asked for a rational expression with a hole, look for a removable discontinuity. Here the numerator x^2−4 factors as (x−2)(x+2). The (x−2) factor cancels with the denominator, leaving x+2 for x ≠ 2. So as x approaches 2, the function behaves like x+2, which tends to 4. Therefore the limit is 4. This is a removable discontinuity at x=2, since you’d get continuity if you defined f(2)=4. The other values would require different near-2 behavior, but cancellation shows the limit is 4.

When a limit is asked for a rational expression with a hole, look for a removable discontinuity. Here the numerator x^2−4 factors as (x−2)(x+2). The (x−2) factor cancels with the denominator, leaving x+2 for x ≠ 2. So as x approaches 2, the function behaves like x+2, which tends to 4. Therefore the limit is 4. This is a removable discontinuity at x=2, since you’d get continuity if you defined f(2)=4. The other values would require different near-2 behavior, but cancellation shows the limit is 4.

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