25 lines
998 B
Plaintext
25 lines
998 B
Plaintext
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{
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"problem": "How many vertical asymptotes does the graph of $y=\\frac{2}{x^2+x-6}$ have?",
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"level": "Level 3",
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"type": "Algebra",
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"solution": "The denominator of the rational function factors into $x^2+x-6=(x-2)(x+3)$. Since the numerator is always nonzero, there is a vertical asymptote whenever the denominator is $0$, which occurs for $x = 2$ and $x = -3$. Therefore, the graph has $\\boxed{2}$ vertical asymptotes."
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}
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theorem:the graph of y=2/(x^2+x-6) has 2 vertical asymptotes.
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Proof.
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Define vertical asymptote as lim_{x->c} f(x) = ∞ or -∞.
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The denominator of the rational function factors into x^2+x-6=(x-2)(x+3).
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Since the numerator is always nonzero, there is a vertical asymptote whenever the denominator is 0,
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which occurs for x = 2 and x = -3.
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Therefore, the graph has 2 vertical asymptotes.
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Qed.
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-/
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import Mathlib.Data.Real.Basic
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-- noncomputable def f (x : ℝ) : ℝ := 2 / (x^2 + x - 6)
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noncomputable def f (x : ℝ) : ℝ := 2 / (x^2 + x - 6)
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#check f
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