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Calculus practice

Evaluating Limits at Infinity

Practice exact positive- and negative-infinity limits for polynomial, rational, trigonometric, exponential, and logarithmic functions.

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Evaluate each limit exactly and explain which asymptotic feature controls the tail.

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Read the tail before calculating

Students compare leading degrees, bounded factors, and exponential or logarithmic growth before reporting an exact finite, infinite, or does-not-exist result.

Learning objective

What this activity builds

Use dominant terms, boundedness, continuity, and growth order to evaluate symbolic limits at infinity exactly.

Worked example

What controls a rational function when the numerator and denominator have equal degree?

The ratio of their leading coefficients.

Lower-order terms vanish after dividing through by the common highest power of x.

Teaching notes

Make the practice count

  • Check the direction before assigning the sign of an odd power.
  • Treat persistent trigonometric oscillation as DNE.
  • Verify a logarithm is eventually defined on the selected tail.

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Math SheetNameScoreTeacherDateEvaluating Limits at InfinityEvaluate each limit exactly. Show which leading term, bounded factor, or growth comparison controls the tail.1.limx-8·x3+9·x2-2·x+7Exact limit:2.limx+3·x2+x+59·x2-2·x-8Exact limit:3.limx-1-3·tan-x+2Exact limit:4.limx+-3+3·3-x-1Exact limit:5.limx+-8·x4-5·x3-2·x2-7·x+1Exact limit:6.limx--8·x4+9·x3-6·x2-9·x-9-8·x-4Exact limit:Exact answers may use pi, ln, or signed infinity.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateEvaluating Limits at InfinityEvaluate each limit exactly. Show which leading term, bounded factor, or growth comparison controls the tail.7.limx+3+12·sin2·x+3x-12Exact limit:8.limx-3-3·x42-2·x+1Exact limit:Exact answers may use pi, ln, or signed infinity.math-sheet.comPage 2 of 2

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