Answer
\[\frac{1}{24}\]
Work Step by Step
\[\begin{align}
& \underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{2}}/2-1+\cos x}{{{x}^{4}}} \\
& \text{Evaluating the limit directly} \\
& \underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{2}}/2-1+\cos x}{{{x}^{4}}}=\frac{{{0}^{2}}/2-1+\cos 0}{{{0}^{4}}}=\frac{0}{0} \\
& \text{The limit has the indeterminate form }\frac{0}{0},\text{ then we can apply} \\
& \text{the L }\!\!'\!\!\text{ Hopital }\!\!'\!\!\text{ s Rule} \\
& \text{The Mclaurin series for }\cos x\text{ is: }\left( \text{go to page 694}\text{, table 9}\text{.5} \right) \\
& \cos x=1-\frac{{{x}^{2}}}{2}+\frac{{{x}^{4}}}{24}-\frac{{{x}^{6}}}{720}+\cdots \\
& \text{Substituting into the given limit} \\
& \underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{2}}/2-1+\cos x}{{{x}^{4}}} \\
& \underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{2}}/2-1+\left( 1-\frac{{{x}^{2}}}{2}+\frac{{{x}^{4}}}{24}-\frac{{{x}^{6}}}{720}+\cdots \right)}{{{x}^{4}}} \\
& \text{Simplifying} \\
& \underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{{{x}^{2}}}{2}-1+1-\frac{{{x}^{2}}}{2}+\frac{{{x}^{4}}}{24}-\frac{{{x}^{6}}}{720}+\cdots }{{{x}^{4}}} \\
& \underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{{{x}^{4}}}{24}-\frac{{{x}^{6}}}{720}+\cdots }{{{x}^{4}}} \\
& \underset{x\to 0}{\mathop{\lim }}\,\left( \frac{1}{24}-\frac{{{x}^{2}}}{720}+\cdots \right) \\
& \text{Evaluating the limit} \\
& =\frac{1}{24}-\frac{{{\left( 0 \right)}^{2}}}{720}+\cdots \\
& =\frac{1}{24} \\
\end{align}\]
