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# 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.$\displaystyle \lim_{x\to 0} \frac{\sinh x - x}{x^3}$

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So here we see if we have signed me Jax my ass acts divided by X cubed. We plug in zero will get 0/0 which is an indeterminate form. So we're going to use the hotel's role in this case and were allowed to um So based on that when we take the derivative of the numerator, we'll get coastline H. X minus one. Um over three square. But we're still going to end up getting 0/0 when we plug in zero. So another ended component form means another round of lope towels role. We do this and we'll end up getting the sign. It acts over six X. And once again when we plug in infinity. Well when we plug in zero will get 0/0. The indeterminate form. So let's try this one more time. This time it will actually work. We'll get close any checks over six. Then when we plug in zero for one or zero for X, we end up getting 1/6. 1/6 is I'm going to be our final answer of the limit, and that was through multiple rounds of look tiles role. But using locales role is still fairly simple compared to other methods.

California Baptist University

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