Question

Use Maclaurin to do problem 26 to 29 and check your results by computer.

$\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathbf{(}\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{-}\mathbf{c}\mathbf{o}{\mathbf{t}}^{\mathbf{2}}\mathbf{x}\mathbf{)}$

Short answer

Answer using Maclaurin theorem is

$\underset{\mathrm{x}\to 0}{\lim }(\frac{1}{{\mathrm{x}}^2}-{\cot }^2\mathrm{x})=\frac{2}{3}$

Step-by-step solution

Given information and L Hospital’s rule:

Given:$\underset{\mathrm{x}\to 0}{\lim }(\frac{1}{{\mathrm{x}}^2}-{\cot }^2\mathrm{x})$

Formula: L Hospital’s rule.

Ifrole="math" localid="1658896015169" $\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}\frac{\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}{\mathbf{g}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}\mathbf{=}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{t}\mathit{e}\mathit{r}\mathit{m}\mathit{i}\mathit{n}\mathit{a}\mathit{t}\mathit{e}\mathbf{}\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathbf{}\mathbf{\{}\frac{\mathbf{0}}{\mathbf{0}}\mathbf{,}\frac{\mathbf{\approx }}{\mathbf{\infty }}\mathbf{,}\mathbf{0}\mathbf{\times }\mathbf{\infty }\mathbf{\}}$ , then limit can be found by differentiating top , bottom functions and applying limit again to get$\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{lim}}\frac{\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}{\mathbf{g}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}$

Taylor series:

role="math" localid="1658896326667" $$\begin{gathered} \mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{f}}^{\mathbf{\left(}\mathbf{n}\mathbf{\right)}}\mathbf{\left(}\mathbf{a}\mathbf{\right)}}{\mathbf{n}\mathbf{!}}\mathbf{(}\mathit{x}\mathbf{-}\mathit{a}{\mathbf{)}}^{\mathbf{n}} \\ \mathbf{}\mathit{f}\mathbf{(}\mathit{a}\mathbf{)}\mathbf{+}{\mathit{f}}^{\mathbf{\text{'}}}\mathbf{(}\mathit{a}\mathbf{\left)}\mathbf{\right(}\mathit{x}\mathbf{-}\mathit{a}\mathbf{)}\mathbf{+}\frac{\mathbf{f}\mathbf{"}\mathbf{\left(}\mathbf{a}\mathbf{\right)}}{\mathbf{2}\mathbf{!}}\mathbf{(}\mathit{x}\mathbf{-}\mathit{a}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{f}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{a}\mathbf{\right)}}{\mathbf{3}\mathbf{!}}\mathbf{(}\mathit{x}\mathbf{-}\mathit{a}{\mathbf{)}}^{\mathbf{3}}\mathbf{+}\mathbf{\cdots } \end{gathered}$$

Maclaurin series is the special case of the Taylor series centered at :

$$\begin{gathered} \mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{f}}^{\mathbf{\left(}\mathbf{n}\mathbf{\right)}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}}{\mathbf{n}\mathbf{!}}{\mathit{x}}^{\mathbf{n}}\mathbf{}\mathbf{}\mathbf{} \\ \mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{+}{\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathit{x}\mathbf{+}\frac{{\mathbf{f}}^{\mathbf{\text{'}}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}}{\mathbf{2}\mathbf{!}}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{f}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}}{\mathbf{3}\mathbf{!}}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{\cdots } \end{gathered}$$

Calculation.