To analyze the end behavior of the function $f(x) = a \cdot b^x - c$, where $a > 0$, $0 \leq b < 1$, and $c$ is a real number, we compute the limits of $f(x)$ as $x \to \infty$ and $x \to -\infty$:

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1. Behavior as $x \to \infty$:

Since $b$ is between 0 and 1, $b^x \to 0$ as $x \to \infty$. Therefore: $f(x) = a \cdot b^x - c \implies f(x) \to -c \text{ as } x \to \infty.$

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2. Behavior as $x \to -\infty$:

When $x \to -\infty$, $b^x \to \infty$, because $b < 1$ and the reciprocal of a small number grows unbounded. Therefore: $f(x) = a \cdot b^x - c \implies f(x) \to \infty \text{ as } x \to -\infty.$

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Final Answer:

The correct description of the end behavior is: $\lim_{x \to \infty} f(x) = -c \quad \text{and} \quad \lim_{x \to -\infty} f(x) = \infty.$

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Do you have further questions, or should we explore more examples? Here are five related questions for practice:

1. What happens if $a < 0$ in the function $f(x) = a \cdot b^x - c$?
2. How does the graph of $f(x) = a \cdot b^x - c$ change when $c$ is increased or decreased?
3. What is the behavior of $f(x) = a \cdot b^x - c$ when $b = 1$?
4. Can the function $f(x) = a \cdot b^x - c$ ever have horizontal asymptotes? If so, where?
5. How would $f(x) = a \cdot b^x - c$ change if $b > 1$ instead of $0 < b < 1$?

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Tip: The exponential base $b$ dictates the growth or decay behavior. For $b < 1$, the function represents exponential decay. For $b > 1$, it represents exponential growth.