Mathematics

Mathematics

Problem

Consider a function pp whose values near x=0x=0 are given in the table.
xx−0.1-0.1−0.01-0.01−0.001-0.0010.0010.0010.010.010.10.1
p(x)p(x)0.95−0.98-0.980.999−1.002-1.0020.97−0.93-0.93
(a) Based on the numerical evidence, decide whether lim⁡x→0p(x)\displaystyle \lim_{x\to 0} p(x) exists. Clearly articulate your conclusion and support it with data from the table. (b) Provide a rigorous argument, written in complete sentences, that explains why the observed oscillatory pattern in the values of p(x)p(x) prevents the limit from existing. (c) A new function qq is defined by q(x)=x2p(x)q(x)=x^2 p(x). Without computing qq analytically, a student generates the following supplementary table for qq near x=0x=0.
xx−0.1-0.1−0.01-0.01−0.001-0.0010.0010.0010.010.010.10.1
q(x)q(x)0.0095−0.000098-0.0000980.000000999−0.000001002-0.0000010020.00000097−0.0093-0.0093
Use the data to estimate lim⁡x→0q(x)\displaystyle \lim_{x\to 0} q(x) and justify why this limit is different from the limit analyzed in part (a).
Intuitive explanation

The key to understanding this function's behavior near x=0x=0 is recognizing the oscillating pattern in the table: as xx approaches 0 from either side, p(x)p(x) alternates between values near +1+1 and −1-1. This tells you the function doesn't settle toward a single limit—instead, it's likely dominated by a rapidly oscillating component (like cos⁡(1/x)\cos(1/x) or sin⁡(1/x)\sin(1/x)) that prevents the limit from existing. Watch how the sign flips as you move closer to 0, and think about what kind of function could produce such wild swings in such a tiny interval.

Intuitive explanation

The key to understanding this function's behavior near x=0x=0 is recognizing the oscillating pattern in the table: as xx approaches 0 from either side, p(x)p(x) alternates between values near +1+1 and −1-1. This tells you the function doesn't settle toward a single limit—instead, it's likely dominated by a rapidly oscillating component (like cos⁡(1/x)\cos(1/x) or sin⁡(1/x)\sin(1/x)) that prevents the limit from existing. Watch how the sign flips as you move closer to 0, and think about what kind of function could produce such wild swings in such a tiny interval.

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