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Title: limit 0/0 form
Description: Proof of an important limit problem to be used in almost all calculus problems in particular differentiation.The strategy of the proof is too simple with detail of all relevant concepts
Description: Proof of an important limit problem to be used in almost all calculus problems in particular differentiation.The strategy of the proof is too simple with detail of all relevant concepts
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Find the indicated lim
ax - 1
x
x→0
,
Since at x = 0
⟹
ax - 1
x
ax = a0 = 1
⟹ ax - 1 = 0
is in
0
0
form also called undefined form
...
In exponential form
ax = e = e1
ax = e1
Dividing both exponents by x
x
ax
=
a=
1
ex
1
ex
OR
1
ex
=a
In log form
1
--------- (7)
x
From equation (6) & equation (7)
log e a =
log e a =
1
log a e
The log with base e is called natural logarthim, and written as ln
in symbolic form
...
{ n - (n - 1)} 1
+
...
+
1
n!
1-
1
1-
2!
1
1-
n
1
n
2
+
1
3!
3
1-
n
n
1-
1
n
2
1-
...
n-1
n
Since
1
1
1
1<
n
2
2!
1
1
2
1
11< 2
3!
n
n
2
and similarly
1
n!
1+
1
n
1-
1
n
1-
n
= 1+1+
1
2!
2
n
...
+
1
n!
1-
1
n
1-
2
n
3
1-
...
1 - 1 +
1
< 1 + (1 +
n
1
2
+
...
+
1
2 n-1
=
1-
1
2n
= 2 1-
1
1- 2
1
= 2-
2n
1
2 n-1
Putting equation (12) in equation (11)
1+
1
n
n
< 1+2-
1
2 n-1
When n approaches infinity
lim 1 +
n→∞
1
n
< lim 3 n→∞
1
2 n-1
Hence 2 < e < 3
<3
-------- (12)
Title: limit 0/0 form
Description: Proof of an important limit problem to be used in almost all calculus problems in particular differentiation.The strategy of the proof is too simple with detail of all relevant concepts
Description: Proof of an important limit problem to be used in almost all calculus problems in particular differentiation.The strategy of the proof is too simple with detail of all relevant concepts