Question

Use the given Maclaurin series to evaluate the limit

Answer 1

The "given series" should be for $\cos(x)$, not $x$, so that

$\cos(x) = 1 - \dfrac{x^2}2 + \dfrac{x^4}{24} - \dfrac{x^6}{720} + \cdots$

In the limit (which should say $x\to\infty$, not $n$), we have

$\displaystyle \lim_{x\to\infty} \frac{\frac{x^2}{1+\cos(x)}}{x^4} = \lim_{x\to\infty} \frac{1}{x^2\left(2 - \frac{x^2}2 + \frac{x^4}{24} - \cdots\right)} = \boxed{0}$