Evaluate the limit.

To evaluate this limit we rewrite it in the indeterminate form and apply L’Hopital’s rule,

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Stumbling Robot

A Fraction of a Dot
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Compute the limit of the given function

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Evaluate the limit.

To evaluate this limit we rewrite it in the indeterminate form and apply L’Hopital’s rule,

So, Apostol explicitly says that he’s not going to introduce forms lof L’Hopital’s rule for expressions of the form infinity/infinity. He says this on the bottom of page 300. So I don’t think it’s in keeping with the spirit of the text to use L’Hopital here.

The solution that I think Apostol may be looking for here is by approximation with a Taylor polynomial. If you substitute x = t + 1, then log(1 – x) = log t, and you can get a nice Taylor polynomial for log x = log (t + 1). Then apply the result from Example 2 on page 302.