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APPLICATIONS OF TAYLOR SERIES
1
Applications of Taylor Series
We have seen how Taylor series can provide a useful approximation of a
function on an interval. Furthermore, we have been able to use Taylor’s
remainder theorem to bound the maximum error between a function and its
Taylor polynomial on an interval.
In fact, there are many further applications of Taylor series—it is one
of the most useful tools in applied mathematics. The reason is simple:
performing many standard mathematical operations (e.g. di±erentiation,
integration, taking limits, etc.) is di²cult for many standard functions;
however, it is very easy to perform these operations to monomial terms.
Since a Taylor Series consists only of terms of this sort, we can often obtain a
very good approximation to the desired solution even when an exact solution
to a problem eludes us. Sometimes we can even recover the exact solution
itself!
1.1
Limits
We can often use Taylor series to simplify improper limits, limits of the form
“0
/
0” or “
∞
/
∞
”. Previously, we have used l’Hopital’s rule—now we will
make use the fact that in the limits
x
→
0 and
x
→ ∞
the monomial terms
in the Taylor series have well de³ned behaviours.
Example 1:
Evaluate the limit
lim
x
→
0
sin(
x
)
x
.
This is a limit of the form “0
/
0”. Previously, we had evaluated this
limit using l’Hopital’s rule; however, since we now know the Taylor series
expansion of sin(
x
) we can use this to simplify the limit.
1
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View Full Documentlim
x
→
0
sin(
x
)
x
= lim
x
→
0
x

x
3
3!
+
x
5
5!
 ···
x
= lim
x
→
0
b
1

x
2
3!
+
x
4
5!
···
B
= 1
.
We recall that by l’Hopital’s rule we had obtained
lim
x
→
0
sin(
x
)
x
= lim
x
→
0
cos(
x
)
1
= 1
.
Example 2:
Evaluate the limit
lim
x
→
0
x
e
x

1
.
This is a limit of the form ”0
/
0”. We use the Taylor series expansion of
e
x
to obtain
lim
x
→
0
x
e
x

1
= lim
x
→
0
x
p
1 +
x
+
x
2
2!
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 Spring '09
 Soares

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