document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179801>Question 1179801</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Arrange the functions for which the result is a non-infinite value and the limit exists in ascending order of their limit values as x tends to infinity. \r<BR>\n" );
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document.write( "Only 6 functions will be used\r<BR>\n" );
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document.write( "f(x)=x^2-1,000/x-5<BR>\n" );
document.write( "j(x)=x^2-1/|7x-1|<BR>\n" );
document.write( "h(x)=x^3-x^2+4/1-3x^3<BR>\n" );
document.write( "i(x)=x-1/|1-4x|<BR>\n" );
document.write( "k(x)=5x+1,000/x^2<BR>\n" );
document.write( "m(x)=- 4x^2-6/1-4x^2<BR>\n" );
document.write( "g(x)=|4x-1|/x-4<BR>\n" );
document.write( "l(x)=5x^2-4/x^2+1\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850170 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850170\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the analysis of the functions and their limits as x approaches infinity:\r<BR>\n" );
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document.write( "**1. Analyze the functions:**\r<BR>\n" );
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document.write( "* **f(x) = (x² - 1000) / (x - 5):**<BR>\n" );
document.write( "    * As x → ∞, the x² term dominates, and the function behaves like x²/x = x.<BR>\n" );
document.write( "    * Therefore, the limit is ∞ (infinite).\r<BR>\n" );
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document.write( "* **j(x) = (x² - 1) / |7x - 1|:**<BR>\n" );
document.write( "    * As x → ∞, the x² term in the numerator and the 7x term in the denominator dominate.<BR>\n" );
document.write( "    * The function behaves like x²/7x = x/7.<BR>\n" );
document.write( "    * Therefore, the limit is ∞ (infinite).\r<BR>\n" );
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document.write( "* **h(x) = (x³ - x² + 4) / (1 - 3x³):**<BR>\n" );
document.write( "    * As x → ∞, the x³ terms dominate.<BR>\n" );
document.write( "    * The function behaves like x³ / (-3x³) = -1/3.<BR>\n" );
document.write( "    * Therefore, the limit is -1/3.\r<BR>\n" );
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document.write( "* **i(x) = (x - 1) / |1 - 4x|:**<BR>\n" );
document.write( "    * As x → ∞, the x terms dominate.<BR>\n" );
document.write( "    * The function behaves like x / |-4x| = x / 4x = 1/4.<BR>\n" );
document.write( "    * Therefore, the limit is 1/4.\r<BR>\n" );
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document.write( "* **k(x) = (5x + 1000) / x²:**<BR>\n" );
document.write( "    * As x → ∞, the x² term in the denominator dominates.<BR>\n" );
document.write( "    * The function behaves like 5x / x² = 5/x.<BR>\n" );
document.write( "    * Therefore, the limit is 0.\r<BR>\n" );
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document.write( "* **m(x) = (-4x² - 6) / (1 - 4x²):**<BR>\n" );
document.write( "    * As x → ∞, the x² terms dominate.<BR>\n" );
document.write( "    * The function behaves like -4x² / (-4x²) = 1.<BR>\n" );
document.write( "    * Therefore, the limit is 1.\r<BR>\n" );
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document.write( "* **g(x) = |4x - 1| / (x - 4):**<BR>\n" );
document.write( "    * As x → ∞, the 4x term in the numerator and the x term in the denominator dominate.<BR>\n" );
document.write( "    * The function behaves like 4x / x = 4.<BR>\n" );
document.write( "    * Therefore, the limit is 4.\r<BR>\n" );
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document.write( "* **l(x) = (5x² - 4) / (x² + 1):**<BR>\n" );
document.write( "    * As x → ∞, the x² terms dominate.<BR>\n" );
document.write( "    * The function behaves like 5x² / x² = 5.<BR>\n" );
document.write( "    * Therefore, the limit is 5.\r<BR>\n" );
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document.write( "**2. Identify the functions with non-infinite limits:**\r<BR>\n" );
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document.write( "The functions with non-infinite limits are h(x), i(x), k(x), m(x), g(x), and l(x).\r<BR>\n" );
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document.write( "**3. Arrange in ascending order:**\r<BR>\n" );
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document.write( "* h(x) = -1/3 ≈ -0.333<BR>\n" );
document.write( "* k(x) = 0<BR>\n" );
document.write( "* i(x) = 1/4 = 0.25<BR>\n" );
document.write( "* m(x) = 1<BR>\n" );
document.write( "* g(x) = 4<BR>\n" );
document.write( "* l(x) = 5\r<BR>\n" );
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document.write( "**Final Answer:**\r<BR>\n" );
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document.write( "The functions in ascending order of their limits as x tends to infinity are: h(x), k(x), i(x), m(x), g(x), l(x).<BR>\n" );
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