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A lab compares two cultures. After x hours the first has 3ˣ million cells and the second has 2ˣ…

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A lab compares two cultures. After $x$ hours the first has $3^x$ million cells and the second has $2^x$ million cells. At $x = 0$ both have $1$ million cells. The scientist studies how fast the difference of the two grows at the start, using the ratio $D(x) = \dfrac{3^x - 2^x}{x}$ million cells per hour. Find the value that $D(x)$ approaches as $x$ tends to $0$ , to three decimal places. Write $3^x - 2^x = (3^x - 1) - (2^x - 1)$ .

Need a reminder? The key ideaShow

With $x$ in radians: $\dfrac{\sin x}{x} \to 1$ and $\dfrac{\tan x}{x} \to 1$ as $x \to 0$ ; $\dfrac{x^n - a^n}{x - a} \to na^{n-1}$ as $x \to a$ ; $\dfrac{a^x - 1}{x} \to \ln a$ as $x \to 0$ .

The worked solution

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