`lim_(x->0) (sinh(x) - x)/(x^3)` 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...

`lim_(x->0) (sinh(x) - x)/(x^3)` 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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Textbook Question

Chapter 4, 4.4 - Problem 26 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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nees101's profile pic

nees101 | Student, Graduate | (Level 2) Adjunct Educator

Posted on

Given the limit function `lim_{x->0}(sinh(x)-x)/x^3`

``

Applying the limits we have,

`lim_{x->0}(sinh(x)-x)/(x^3)=0/0`

Using L'Hospital's rule we get,

`lim_{x->0}(cosh(x)-1)/(3x^2)=0/0`

So, again using L'Hospital's rule we have,

`lim_{x->0}(sinh(x))/(6x)=0/0`

Again applying L'Hospital's rule we get,

`lim_{x->0}(cosh(x))/6=1/6`

hence the limit is `1/6`

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scisser | (Level 3) Honors

Posted on

Apply LH's rule

`=lim_(x-gtoo)((cosh-1)/(3x^2))=oo/oo`

Apply LH's rule again

=`lim_(x->oo)((sinhx)/6x)=oo/oo `

Again, LH's rule

`=lim_(x-gtoo)((coshx)/6)`

`=1/6`

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