Dear Uncle Colin,

I need to find the limit as $x$ approaches 1 of $\frac{x^{29}-1}{x-1}$. I tried factoring out $x^{28}$ but didn’t get anywhere.

– Learning How Others Proceed In This Awful Limit

Hi, LHOPITAL, and thanks for your message!

Factoring out an $x^{28}$ is very unlikely to get you anywhere, on the grounds that there isn’t a factor of $x^{28}$! However, there are several excellent alternative approaches.

Long division

The first, and perhaps most obvious, approach would be to divide the top by the bottom and see what shakes loose. That’s quite a tedious thing, but after…

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