Chapter 11: Problem 13

Determine the order, \(O\left(x^{p}\right)\), of the following functions. You may need to use series expansions in powers of \(x\) when \(x \rightarrow 0\), or series expansions in powers of \(1 / x\) when \(x \rightarrow \infty\) a. \(\sqrt{x(1-x)}\) as \(x \rightarrow 0\). b. \(\frac{x^{5 / 4}}{1-\cos x}\) as \(x \rightarrow 0\) c. \(\frac{x}{x^{2}-1}\) as \(x \rightarrow \infty\). d. \(\sqrt{x^{2}+x}-x\) as \(x \rightarrow \infty\).

Short Answer

Expert verified

a. \textit{O}\(x^{1}\), b. \textit{O}\(x^{1/4}\), c. \textit{O}\(x^{-1}\), d. \textit{O}(\(x^{-\infty}\)).

Step by step solution

(Problem a)

For the function \(\sqrt{x(1-x)}\) as \(x \rightarrow 0\), the function can be simplified as \(x \rightarrow 0\), to \(x*1\). When finding the order of this function as \(x \rightarrow 0\), \textit{O}\(x^{1}\), the function simplifies to \(\sqrt{x*1}\) = \(x\).\n

(Problem b)

For the function \(\frac{x^{5 / 4}}{1-\cos x} \) as \(x \rightarrow 0\), apply the series expansion of \(cos(x)\). This gives us \(\frac{x^{5 / 4}}{1- (1 - x^2/2)}\), which further simplifies to \(\frac{x^{5 / 4}}{x^2 / 2}\) = \(x^{1/4}\). So the function is \textit{O}\(x^{1/4}\) as \(x \rightarrow 0\).\n

(Problem c)

For the function \(\frac{x}{x^{2}-1}\) as \(x \rightarrow \infty\), expand the function in powers of \(1/x\) which gives us \(\frac{1} {x-1/x}\) and as \(x \rightarrow \infty\) this simplifies to \(\frac{1}{x}\). Hence, the function is \textit{O}\(x^{-1}\).\n

(Problem d)

For the function \(\sqrt{x^{2}+x}-x\) as \(x \rightarrow \infty\), factor \(x\) out first which gives us \(\sqrt{x^{2}(1+1/x)}-x\). This simplifies to \(x\sqrt{1+1/x}-x\) and once \(x \rightarrow \infty\), this works out to \(x-x\)=0. That implies, the function is \textit{O}(\(x^{-\infty}\)).\n