Limits and comparative growth
Limits at infinity and at a point, operations on limits, the four indeterminate forms, horizontal and vertical asymptotes, and the comparative growth of logarithms, powers and exponentials.
- 25 cards
- 6 topics
- Free
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Each topic is weighted the way the exam weighs it.
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A few cards from the deck
Limit at +∞: what is it?
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A number ℓ that f(x) gets as close to as we like when x becomes large enough; or +∞ if f(x) eventually exceeds any threshold, −∞ if it eventually drops below any threshold.
1/x tends to 0 at +∞, x² tends to +∞, and cos x has no limit: it oscillates between −1 and 1 without settling.
What is the limit of (3x² + 1) / (x² − 4) as x tends to +∞?
- 3
- +∞
- 0
- 1
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3
Numerator and denominator tend to +∞: form ∞/∞. Factoring x² top and bottom gives (3 + 1/x²) / (1 − 4/x²), which tends to 3/1 = 3. Only the highest-degree terms count.
The four indeterminate forms are +∞ − ∞, …, … and 0 / 0.
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The four indeterminate forms are +∞ − ∞, 0 × ∞, ∞ / ∞ and 0 / 0.
The other operations with infinities conclude directly: (+∞) + (+∞) = +∞, (+∞) × (−∞) = −∞, 5 / (+∞) = 0. 1 / 0⁺ = +∞ also concludes, since the sign of the denominator is known.
Indeterminate form: what is it?
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A notation such as +∞ − ∞ that does not let us conclude: depending on the functions involved, the limit can be 0, a real number, +∞, −∞, or not exist.
x − x tends to 0, x² − x to +∞ and x − x² to −∞, though each term tends to +∞: the expression must be transformed (factored, simplified) before concluding.