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Example 5
Show that lim
x→0
x
2
sin
1
x
=0.
The Euler number e has been earlier introduced as a limit of a
certain sequence. Let us notice that the same number can be defined
as either lim
x→∞
1+
1
x
x
or lim
y→0
(1 + y)
1/y
.
Here is an example how to use it when dealing with certain
indeterminate expressions.
Example 6
Let f(x)=
1+
3
x
x
. It is an indeterminate expression of the
form 1
∞
when x →∞. To deal with it, introduce y =3/x. Clearly,
y → 0 as x →∞.Now,
lim
x→∞
1+
3
x
x
= lim
y→0
(1 + y)
3/y
=
lim
y→0
(1 + y)
1/y
3
= e
3
.
Here are a few more limits to remember:
lim
x→0
sin x
x
=1,
lim
x→0
log
α
(1 + x)
x
=log
α
e =
1
ln α
,
lim
x→0
(1 + x)
α
− 1
x
= α,
lim
x→0
α
x
− 1
x
=lnα.
3. Continuity
You can notice that the limit of a function as x approaches a can
often be found simply by calculating the value of the function at a.
Functions with this property are called continuous at a. We will see
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