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A rational function is undefined at zeros of its original denominator.
Worked example 1
01 / For r(x)=−2−x−12, complete a table at inputs 109, 10099, 100101, 1011, −100, 100. Use it to describe both one-sided behaviors at 1 and predict the horizontal asymptote. Explain why finite table values support, but do not prove, an infinite limit. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
02 / Substitute each input without rounding: 18, 198, −202, −22, −101200, −99200.
03 / The factor x−1 changes sign across the excluded input.
04 / Thus the left and right limits are ∞ and −∞.
05 / As the absolute input grows, the fractional term tends to zero; the exact end limit is −2.
06 / A finite table samples only finitely many inputs and finite outputs. It cannot establish what happens at every input arbitrarily close to the excluded value, or prove that the outputs exceed every bound. Here the algebra supplies that proof: the nonzero constant numerator divided by a denominator approaching zero has unbounded magnitude, while the denominator sign determines the one-sided direction.
Worked example 2 · Use tables to distinguish one-sided and end behavior
For r(x)=−1+(x+2)23, complete a table at inputs −1021, −100201, −100199, −1019, −100, 100. Use it to describe both one-sided behaviors at −2 and predict the horizontal asymptote. Explain why finite table values support, but do not prove, an infinite limit. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Substitute each input without rounding: 299, 29999, 29999, 299, −96049601, −34683467. The factor (x+2)2 is positive on both sides of the excluded input. Thus the left and right limits are ∞ and ∞. As the absolute input grows, the fractional term tends to zero; the exact end limit is −1. A finite table samples only finitely many inputs and finite outputs. It cannot establish what happens at every input arbitrarily close to the excluded value, or prove that the outputs exceed every bound. Here the algebra supplies that proof: the nonzero constant numerator divided by a denominator approaching zero has unbounded magnitude, while the denominator sign determines the one-sided direction. Written-work review:
Show the stated algebra and preserve every excluded input.
Include all requested graph, sign-table, model or explanation components.
Compare the written work with the supplied reference solution; selecting a statement alone does not establish mastery.
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.