problem_11

Author: eigenlambda123

01_limits/1.9_continuity/problem_11.ipynb

@@ -15,12 +15,32 @@
    ]
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-   "execution_count": null,
-   "id": "e4615d78-9088-47de-ab90-97ea4749e764",
+   "cell_type": "markdown",
+   "id": "a9339760-cf1c-4872-8129-0284f6c20d71",
    "metadata": {},
-   "outputs": [],
-   "source": []
+   "source": [
+    "---\n",
+    "\n",
+    "### **Solution**\n",
+    "\n",
+    "A rational function is discontinuous where its denominator is zero.\n",
+    "\n",
+    "**Find where the denominator is zero:**\n",
+    "Set the denominator equal to zero:\n",
+    "$$7 - e^{2x+3} = 0$$\n",
+    "\n",
+    "Solve for the exponential term:\n",
+    "$$e^{2x+3} = 7$$\n",
+    "\n",
+    "**Solve for $x$:**\n",
+    "Take the natural logarithm of both sides:\n",
+    "$$2x + 3 = \\ln(7)$$\n",
+    "$$2x = \\ln(7) - 3$$\n",
+    "$$x = \\frac{\\ln(7) - 3}{2}$$\n",
+    "\n",
+    "**Conclusion:**\n",
+    "The function $h(x)$ is discontinuous at $x = \\frac{\\ln(7) - 3}{2}$."
+   ]
   }
  ],
  "metadata": {