APFIVE Error Analysis in Limit Evaluation
A student investigates the limit
$$ \lim_{x \to 0} \frac{\sqrt{9 + x} - 3}{x} $$
by directly canceling terms, and arrives at an answer of infinity. What mistake did they make?
A
They should have applied L’Hôpital’s rule, which would yield a finite limit.
B
They incorrectly canceled the square root term without rationalizing the numerator, which is necessary to properly evaluate the limit.
C
They substituted $$ x = 0 $$ prematurely.
D
They made a mistake in computing the value of the square root at 9.
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