Determining Limits Using the Squeeze Theorem AB & BC
If $g(x) \le f(x) \le h(x)$ near $a$, and $g$ and $h$ share the same limit $L$ there, then $f$ is trapped and its limit is also $L$. The theorem's value is that $f$ itself need not be tractable: the standard case is a bounded but limitless factor such as $\sin\left(\frac{1}{x}\right)$ multiplied by something shrinking to zero, where the limit laws cannot be applied at all.
The failures are all failures to verify. Bounds get quoted without being multiplied through, so $-1 \le \sin \le 1$ is offered as though it settled a limit when those bounds do not meet. Bounds that close on different values get used anyway, and the theorem concludes nothing there. And bounds valid only for $x > 0$ get applied to a two-sided limit, which matters because multiplying an inequality by a negative quantity reverses it.
The work
3 ways in · any order
Lesson
Determining Limits Using the Squeeze Theorem
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States both hypotheses and why each is needed, builds the bounds for the standard oscillating example step by step, and handles the sign problem when the multiplier is not always positive.
Diagnostic
10-item topic check
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Ten items on the single failure mode of this topic: invoking the Squeeze Theorem without verifying that the bounds hold near the point and share a limit. Take it cold to see whether it is yours, or after the lesson to confirm it is not.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.