`lim_(x->0) (cos(x) - 1 + (1/2)x^2)/(x^4)` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule...

`lim_(x->0) (cos(x) - 1 + (1/2)x^2)/(x^4)` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

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Chapter 4, 4.4 - Problem 39 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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nees101 | Student, Graduate | (Level 2) Adjunct Educator

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Given limit function is: `lim_{x->0}(cos(x)-1+(1/2)x^2)/x^4`  

We have to find the limits.

Applying the limit we see that,

`lim_{x->0}(cos(x)-1+(1/2)x^2)/x^4=0/0`

So now applying the L'Hospital's rule and then applying the limit we get,

`lim_{x->0}(-sin(x)+x)/(4x^3)=0/0`

So again we have to apply L'Hospital's rule i.e.

`lim_{x->0}(-cos(x)+1)/(12x^2)=0/0`

Again applying L'Hospital's rule we get,

`lim_{x->0}sin(x)/(24x)=1/24`    , Since we know that `lim_{x->0}sin(x)/x=1`

 Hence the limit is `1/24`

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