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Example Problem. #8, Lesson 4.5
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Find the following limit. Use l'Hopital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hopital's Rule doesn't apply, explain why.
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Given Problem: #8, Lesson 4.5 |
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First, we examine whether l'Hopital's Rule is appropriate.
We see that this limit is of the indeterminate form:  ,
therefore l'Hopital's Rule is appropriate. |
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Now we apply l'Hopital's Rule and see if this leads us to the limit. And it does. We have found the limit. |
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We look at a graph of the function  ,
especially what happens to it when x approaches zero. We see that
the limit of 2 looks correct.. |
Example Problem. #22, Lesson 4.5
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Find the following limit. Use l'Hopital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hopital's Rule doesn't apply, explain why.
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Given Problem: #22, Lesson 4.5 |
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First, we examine whether l'Hopital's Rule is appropriate.
We see that this limit is of the indeterminate form: ,
therefore l'Hopital's Rule is appropriate. |
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Now we apply l'Hopital's Rule and see if this leads us to the limit. |
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We see that it did not lead us to the limit. So we may wish to use l'Hopital again. |
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Now, we can actually find the limit. And the limit is zero. |
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Looking at the graph of the function at the left, and the closer look below, we see that the limit does appear to be correct. The graph at the left is graphed over the interval (-10,2) |
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This is the same graph over the interval (-29,-20). |
Example Problem. #28, Lesson 4.5
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Find the following limit. Use l'Hopital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hopital's Rule doesn't apply, explain why.
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Given Problem. #28, Lesson 4.5 This is of course impossible to calculate as it now stands. Both individual parts of this limit are undefined. |
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Here we use some trigonometric identities, and some algebra to put the expression into a different form. Now we have one of the indeterminate forms that l'Hopital works on. |
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Now we apply l'Hopital's Rule, and we are able to calculate the limit. Below, we again check visually on a graph. |
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The blue curve is the final curve, which I thought looked suspiciously loke a tangent curve. So I added the two curves which summed give us the answer curve [blue], namely csc(x) [green] and -cot(x) [yellow]. For comparison purposes I also graphed the tangent function in purple. You can see that the blue curve does have a limit of zero as x approaches zero. |
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1-27, Odds
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