If numerator is zero, then limit is infinite?

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When evaluating limits, if the numerator is zero and the denominator is non-zero, the limit approaches zero, as shown by lim x-> a (0/x) = 0 and lim x-> +/- infinity (0/x) = 0. However, if both the numerator and denominator approach zero, the limit becomes indeterminate, potentially resulting in any real number or infinity. This distinction is crucial in calculus when analyzing limits involving rational expressions. Understanding these principles helps clarify the behavior of functions as they approach specific points or infinity. Thus, limits involving zero in the numerator require careful consideration of the denominator's behavior.
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lim x-> a (0/x) = 0

and

lim x-> +/- infinity (0/x) = 0

Since 0/x = 0 for all nonzero x
 
thanks
 
JG89 said:
lim x-> a (0/x) = 0

and

lim x-> +/- infinity (0/x) = 0

Since 0/x = 0 for all nonzero x

But, if both the numerator and denominator are approaching zero, then all bets are off. The limit of this rational expression could be any real number or infinite.
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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