Evaluate lim_{x->4} (√x - 2)/(x-4).

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Multiple Choice

Evaluate lim_{x->4} (√x - 2)/(x-4).

Explanation:
When a limit gives a 0/0 form, look for a way to cancel a common factor through algebraic simplification. Here you can rationalize the numerator by multiplying top and bottom by the conjugate √x + 2. This yields (√x − 2)(√x + 2) / [(x − 4)(√x + 2)] = (x − 4)/[(x − 4)(√x + 2)]. For x not equal to 4, the x − 4 factors cancel, leaving 1/(√x + 2). Then as x approaches 4, √x approaches 2, so the limit is 1/(2 + 2) = 1/4.

When a limit gives a 0/0 form, look for a way to cancel a common factor through algebraic simplification. Here you can rationalize the numerator by multiplying top and bottom by the conjugate √x + 2. This yields (√x − 2)(√x + 2) / [(x − 4)(√x + 2)] = (x − 4)/[(x − 4)(√x + 2)]. For x not equal to 4, the x − 4 factors cancel, leaving 1/(√x + 2). Then as x approaches 4, √x approaches 2, so the limit is 1/(2 + 2) = 1/4.

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