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\noindent\underline{Honors HW2} (due 28 January) \\
Why is it true that $\displaystyle\int_1^x \dfrac{1}{t} \mathrm{d}t = \ln(x)$? In this homework, you will investigate this. \\
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\item Let $F(r) = \displaystyle\int_1^2 t^r \mathrm{d}t$. Calculate $F(r)$ for three values of $r$ close to (but not equal to) $-1$. Do the values of $F(r)$ seem to approach a limit? Do you recognize said limit (if it exists)?
\item Let $x$ be some positive number and asssume $r \neq -1$. Define $G(r)=\displaystyle\int_1^x t^r \mathrm{d}t$ and find a simpler formula for it (i.e. calculate the integral). Based on your formula, why is there trouble when setting $r=-1$?
\item Compute $\displaystyle\lim_{r \rightarrow -1} G(r)$ (hint: L'H\^{o}pital's rule). What do you conclude about how we should interpret $G(-1)$?
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