Connecting Infinite Limits and Vertical Asymptotes AB & BC
An infinite limit is a limit that fails: the outputs grow beyond every bound rather than approaching a real number, and the notation $\lim_{x\to a} f(x) = \infty$ describes the manner of failure. Geometrically, a vertical asymptote at $x = a$ is the statement that at least one one-sided limit there is infinite. Each side needs its own sign analysis, since the two frequently run in opposite directions; an even power in the denominator is what makes them agree.
Two errors dominate. Infinite limits get reported as existing, so "the limit is infinity" is offered as a value, or a single infinite statement is given where the left side falls and the right side rises. And every zero of a denominator gets read as a vertical asymptote, when a factor that cancels against the numerator leaves a hole instead. Factoring, cancelling, and then reading the reduced denominator is what separates the two cases.
The work
3 ways in · any order
Lesson
Connecting Infinite Limits and Vertical Asymptotes
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What infinite notation actually claims, a sign procedure for getting each side's direction, and the factoring check that decides whether an asymptote is there at all.
Diagnostic
10-item topic check
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Ten items spanning the two failure modes of this topic: reporting an infinite limit as an existing one, and calling every denominator zero an asymptote. Take it cold to find which one 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.