Rational Functions and Vertical Asymptotes
A vertical asymptote is unbounded behavior: outputs running to positive or negative infinity as the input approaches a value missing from the domain. Denominator zeros are the suspects, but not all of them blow up. When the numerator shares the zero, the multiplicity comparison decides: if the denominator holds more copies of the factor, the break is an asymptote; if the numerator holds at least as many, the break collapses to a hole. Cancelling changes the kind of break, never erases it.
The course also asks for the blow-up in one-sided limit language: as x approaches the asymptote from the left or right, the outputs increase or decrease without bound, and each side earns its own sign. The traps here are counting an asymptote as a zero, mistaking a hole for an asymptote, and trusting cancellation to make the problem disappear. All three are drilled in the lesson.
The work
3 ways in · any order
Lesson
Rational Functions and Vertical Asymptotes
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Denominator zero plus surviving factor equals blow-up. The lesson runs the multiplicity test that separates asymptotes from holes, reads one-sided limits for the direction of the blow-up, and closes with ten scenarios of inventory-taking on factored rational functions.
Diagnostic
10-item topic check
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Ten items spanning the Topic 1.9 misconceptions: promoting numerator zeros to asymptotes, calling a cancelled factor a hole when a copy survives downstairs, and misreading one-sided unbounded behavior. The bank ships with the Unit 1 data drop.
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.