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Error Analysis in Limit Evaluation

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When evaluating the limit lim⁡x→09+x−3x \lim_{x \to 0} \frac{\sqrt{9 + x} - 3}{x} , a student incorrectly concludes that the limit is infinity. What was the likely error in the student’s procedure?

A

They incorrectly canceled the square root term without rationalizing the numerator, which is necessary to properly evaluate the limit.

B

They substituted x=0 x = 0 prematurely.

C

They made a mistake in computing the value of the square root at 9.

D

They should have applied L’Hôpital’s rule, which would yield a finite limit.

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