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 - \frac{x^2}2 + (x^4)/(24) - (x^6)/(720) + \cdots`

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

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