A sedan will drive ten miles north, 15 miles east, 13 miles south, 15 miles west, and 22 miles north. The sedan gets 20 miles pe
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<span>Let t=time Doreen travels; let r=rate of Doreen
RATE * TIME = DIST.
Sue | r+13 t-3 105
Doreen | r t 64
Now solve the system:
(r+13)(t-3)=105
</span>r*t=64<span>
</span>
<span>
Doreen: rate = 8 mph, time = 8 hours
Sue: </span>rate = 21 mph, time = 5 hours<span>
</span>
RATE * TIME = DIST.
Sue | r+13 t-3 105
Doreen | r t 64
Now solve the system:
(r+13)(t-3)=105
</span>r*t=64<span>
</span>
Doreen: rate = 8 mph, time = 8 hours
Sue: </span>rate = 21 mph, time = 5 hours<span>
</span>
Looks like the given limit is
With some simple algebra, we can rewrite
then distribute the limit over the product,
The first limit is 0, since 1/3ⁿ is a positive, decreasing sequence. But before claiming the overall limit is also 0, we need to show that the second limit is also finite.
For the second limit, recall the definition of the constant, <em>e</em> :
To make our limit resemble this one more closely, make a substitution; replace 9/(<em>n</em> - 9) with 1/<em>m</em>, so that
From the relation 9<em>m</em> = <em>n</em> - 9, we see that <em>m</em> also approaches infinity as <em>n</em> approaches infinity. So, the second limit is rewritten as
Now we apply some more properties of multiplication and limits:
So, the overall limit is indeed 0: