# Use series to evaluate the limit #\lim_(x\rarr0)(x^2/2-1-\cos(x))/x^4#?

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It says #x^2# /#2# in my worksheet. Not sure if it's supposed to be formatted the way it did above.

Anyway... how do I go about doing this?

It says

Anyway... how do I go about doing this?

##### 2 Answers

Using

#### Explanation:

Recall that

#L = lim_(x->0) (x^2/2 - 1 - (1 - x^2/2 + x^4/(4!) + ...))/x^4#

This clearly evaluates to nothing but

#L = lim_(x->0) (x^2/2 - 1 + 1 - x^2/2 + x^4/(4!) + ...)/x^4#

#L = lim_(x->0) (x^4/(4!) - x^6/(6!) + x^8/(8!) + ....)/x^4#

#L = 1/(4!) + 0 + 0 + 0 + ...#

#L = 1/24#

Hopefully this helps!

#### Explanation:

Recall the Maclaurin Series expansion for cosine,

We can insert this series into our limit (that is, the first few terms of the series, followed by the ellipsis to indicate that it goes on forever):

And distribute the negative sign, allowing us to combine the relevant terms:

Divide all terms by

We see that

So, since all of the subsequent terms vanish, we won't include ellipsis anymore as we're no longer dealing with them.

Perform some simplification on the first two terms to take the limit:

Take the limit, and note that the constant