$6$ pts$.)$ Computational scientists use different strategies to avoid pitfalls associated with roundoff error. In your text, the authors demonstrate how rewriting a rational expression can remove problems with catastrophic cancellation. Another strategy is to reduce the number of operations to avoid the accumulation of roundoff error.

In this problem, we attempt both of these.

Repeat the analysis in the second example in Section $2\mathrm{.}1.$ Find

using L'Hopital's rule and using the strategy outlined in the example. You can alter the code below to compute the value of the function for numbers approaching $0$ from the right.