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Chapter 4: Problem 27
In Exercises \(27-30,\) find the limit of \(s(n)\) as \(n \rightarrow \infty\) $$ s(n)=\frac{81}{n^{4}}\left[\frac{n^{2}(n+1)^{2}}{4}\right] $$
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Use the equation of the tractrix \(y=a \operatorname{sech}^{-1} \frac{x}{a}-\sqrt{a^{2}-x^{2}}, \quad a>0\) Let \(L\) be the tangent line to the tractrix at the point \(P .\) If \(L\) intersects the \(y\) -axis at the point \(Q\), show that the distance between \(P\) and \(Q\) is \(a\).
Verify each rule by differentiating. Let \(a>0\). $$ \int \frac{d u}{a^{2}+u^{2}}=\frac{1}{a} \arctan \frac{u}{a}+C $$
Prove that \(\frac{d}{d x}\left[\int_{u(x)}^{v(x)} f(t) d t\right]=f(v(x)) v^{\prime}(x)-f(u(x)) u^{\prime}(x)\).
Find the derivative of the function. \(y=\tanh ^{-1} \frac{x}{2}\)
Verify each rule by differentiating. Let \(a>0\). $$ \int \frac{d u}{u \sqrt{u^{2}-a^{2}}}=\frac{1}{a} \operatorname{arcsec} \frac{|u|}{a}+C $$
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