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A designer studies the ratio E(x) = (x⁵ - 243)/(x² - 9) for an input x in volts (V)…

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A designer studies the ratio $E(x) = \dfrac{x^5 - 243}{x^2 - 9}$ for an input $x$ in volts (V). At $x = 3$ V both the top and the bottom are zero. Find the value that $E(x)$ approaches as $x$ tends to $3$ V. Use the power limit $\dfrac{x^n - a^n}{x - a} \to na^{n-1}$ for the top, and factorise the bottom. Give the answer as a number.

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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