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A sensor reports a ratio Q(x) = (√(x + 6) - 3)/(x - 3) for an input x in volts (V), with x > -6 and…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A sensor reports a ratio $Q(x) = \dfrac{\sqrt{x + 6} - 3}{x - 3}$ for an input $x$ in volts (V), with $x > -6$ and $x \ne 3$ . At $x = 3$ V the top and the bottom are both zero, so the sensor shows nothing. Find the value that $Q(x)$ approaches as $x$ tends to $3$ V, to four decimal places. Use the conjugate of the top, which is $\sqrt{x + 6} + 3$ .

Need a reminder? The key ideaShow

When substitution gives $\dfrac{0}{0}$ , factorise or rationalise, cancel the common factor, and substitute again. For $x \to \infty$ divide the top and the bottom by the highest power of $x$ , and use $\dfrac{1}{x} \to 0$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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