Analysis in Mechanical Engineering |
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© Leon van Dommelen |
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Subsections
4.2 Example
From [1, p. 227, 10v]
Asked:
4.2.1 Observations
You must first look whether the limit is trivial:
Since the product of infinity times zero is unknown, this limit is
nontrivial.
4.2.2 L'Hopital
L'Hopital can be used if you create a ratio of quantities that both
become zero or both become infinite. (For example, you would not want
to apply L'Hopital on .)
Now both top and bottom become infinite. So L'Hopital can be
applied, by differentiating top and bottom separately:
Still infinity over infinity, so differentiate once more
4.2.3 Better
Using some insight is always better than just crunching it out.
First simplify things for yourself by defining . Then
goes to plus infinity instead of minus infinity like .
Then
and that is zero because is much greater than any power of
for large positive . (To see that, just look at the
Taylor series:
The term is much larger than for large and the
other terms make larger still.)
So you could replace by and the limit would still be
zero.