Limit Exponential 34831C

1. **State the problem:** We are given the function $$y = 3 - 2e^{-2x}$$ and asked two questions:\n\n(a) What happens to $y$ as $x \to \infty$?\n(b) Sketch the graph of the function.\n\n2. **Analyze the behavior as $x$ becomes very large:**\n\nRecall that the exponential function $e^{-2x}$ decreases to 0 as $x \to \infty$ because the exponent $-2x$ becomes very large negative.\n\nSo, $$\lim_{x \to \infty} e^{-2x} = 0.$$\n\nTherefore, $$\lim_{x \to \infty} y = 3 - 2 \times 0 = 3.$$\n\nThis means as $x$ becomes very large, $y$ approaches 3.\n\n3. **Sketching the graph:**\n\n- The function is $$y = 3 - 2e^{-2x}.$$\n- When $x=0$, $$y = 3 - 2e^0 = 3 - 2 = 1.$$\n- As $x \to \infty$, $y \to 3$ from below because $-2e^{-2x}$ is negative but approaches 0.\n- As $x \to -\infty$, $e^{-2x} = e^{+\infty} \to \infty$, so $$y \to 3 - 2 \times \infty = -\infty.$$\n\n4. **Summary:**\n\n- The graph starts very low (negative infinity) as $x \to -\infty$.\n- At $x=0$, $y=1$.\n- The graph increases and approaches the horizontal asymptote $y=3$ as $x \to \infty$.\n\n5. **Final answers:**\n\n(a) $y$ approaches 3 as $x$ becomes very large.\n(b) The graph is an increasing curve starting from negative infinity on the left, passing through $(0,1)$, and approaching the horizontal asymptote $y=3$ on the right.