Eagle-eyed friend of the blog @robjlow spotted an error in Uncle Colin’s last answer. As I’m forever telling my students, making errors is how you learn; Rob has graciously delivered a lesson for us all. Thanks for keeping me honest!

Recently, Uncle Colin gave a couple of ways to see that

\[ \lim_{x \to \infty} \left\{ \sqrt{x^2 + 3x} – x\right\} = \frac{3}{2} \]

One approach, the sweet, used the binomial theorem, to write

\[ \sqrt{x^2 + 3x} = x\sqrt{1+\frac{3}{x}} \approx x\left(1+\frac{3}{2x}\right). \]

The very suspicious minded mind be a little hesitant, because the error in…

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