probem_10-12

Author: eigenlambda123

01_limits/1.8_limits_at_infinity_part_2/probem_10-12.ipynb

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+    "### **Problems**\n",
+    "\n",
+    "$$10. \\text{Evaluate } \\lim_{x \\to -\\infty } { tan^{-1}(7-x+3x^{5})}$$\n",
+    "\n",
+    "\n",
+    "$$11. \\text{Evaluate } \\lim_{x \\to \\infty } { tan^{-1}(\\frac{4+7t}{2-t})}$$\n",
+    "\n",
+    "\n",
+    "$$12. \\text{Evaluate } \\lim_{x \\to \\infty } { tan^{-1}(\\frac{3w^{2}-9w^{4}}{4w-w^{3}})}$$"
+   ]
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+    "---\n",
+    "\n",
+    "### **Solutions** \n",
+    "\n",
+    "#### **10.** Given the limit:\n",
+    "$$ \\lim_{x \\to -\\infty} \\tan^{-1}(7 - x + 3x^5) $$\n",
+    "\n",
+    "We analyze the behavior of the argument inside the arctangent as $x \\to -\\infty$:\n",
+    "\n",
+    "- As $x \\to -\\infty$:\n",
+    "  - The term $3x^5$ dominates (since $x^5 \\to -\\infty$ for $x \\to -\\infty$)\n",
+    "  - So $7 - x + 3x^5 \\to -\\infty$\n",
+    "\n",
+    "Therefore:\n",
+    "$$ \\boxed{\\lim_{x \\to -\\infty} \\tan^{-1}(7 - x + 3x^5) = \\tan^{-1}(-\\infty) = -\\frac{\\pi}{2}} $$\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "#### **11.** Given the limit:\n",
+    "$$ \\lim_{t \\to \\infty} \\tan^{-1}\\left(\\frac{4 + 7t}{2 - t}\\right) $$\n",
+    "\n",
+    "We analyze the behavior of the rational expression inside the arctangent as $t \\to \\infty$:\n",
+    "\n",
+    "Divide numerator and denominator by $t$:\n",
+    "$$\n",
+    "\\frac{4 + 7t}{2 - t} = \\frac{\\frac{4}{t} + 7}{\\frac{2}{t} - 1}\n",
+    "$$\n",
+    "\n",
+    "As $t \\to \\infty$:\n",
+    "- $\\frac{4}{t} \\to 0$\n",
+    "- $\\frac{2}{t} \\to 0$\n",
+    "- So the expression $\\to \\frac{0 + 7}{0 - 1} = -7$\n",
+    "\n",
+    "Therefore:\n",
+    "$$ \\boxed{\\lim_{t \\to \\infty} \\tan^{-1}\\left(\\frac{4 + 7t}{2 - t}\\right) = \\tan^{-1}(-7)} $$\n",
+    "\n",
+    "Since $\\tan^{-1}(-7)$ is a finite number (approximately $-1.4289$ radians):\n",
+    "\n",
+    "\n",
+    "#### **12.** Given the limit:\n",
+    "$$ \\lim_{w \\to \\infty} \\tan^{-1}\\left(\\frac{3w^2 - 9w^4}{4w - w^3}\\right) $$\n",
+    "\n",
+    "We analyze the behavior of the rational expression inside the arctangent as $w \\to \\infty$:\n",
+    "\n",
+    "Divide numerator and denominator by $w^3$ (the highest power in the denominator):\n",
+    "$$\n",
+    "\\frac{3w^2 - 9w^4}{4w - w^3} = \\frac{\\frac{3w^2}{w^3} - \\frac{9w^4}{w^3}}{\\frac{4w}{w^3} - \\frac{w^3}{w^3}} = \\frac{\\frac{3}{w} - 9w}{\\frac{4}{w^2} - 1}\n",
+    "$$\n",
+    "\n",
+    "As $w \\to \\infty$:\n",
+    "- $\\frac{3}{w} \\to 0$\n",
+    "- $9w \\to \\infty$\n",
+    "- $\\frac{4}{w^2} \\to 0$\n",
+    "- So the numerator $\\to -\\infty$, denominator $\\to -1$\n",
+    "\n",
+    "Thus:\n",
+    "$$ \\frac{3w^2 - 9w^4}{4w - w^3} \\to \\frac{-\\infty}{-1} = \\infty $$\n",
+    "\n",
+    "Therefore:\n",
+    "$$ \\boxed{\\lim_{w \\to \\infty} \\tan^{-1}\\left(\\frac{3w^2 - 9w^4}{4w - w^3}\\right) = \\tan^{-1}(\\infty) = \\frac{\\pi}{2}} $$\n"
+   ]
+  }
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