- #1
Alpharup
- 225
- 17
Consider the limit
lim f(x)g(x)
x→a
Spivak has proved that this is equal to lim f(x) multlied by
x→a
lim g(x)
x→a
And also if lim g(x) = k and k≠0,
x→a
Then. lim 1/g(x) = 1/k
x→a
Now the problem arises...
Consider the limit
lim ((x^2)-(a^2))/(x-a)
x→a
It can factorised and written as( taking x-2 from numerator)
lim (x+a)
x→a
Which is nothing but 2a.
Now we can write it the above limit also as
lim(x^2)-(a^2) multiplied by
x→a
lim 1/(x-a)
x→a.
The second limit does not exist because
lim(x-a)=0 and l=0
x→a
So, its reciprocal limit does not exist.
Then can't we say
lim ((x^2)-(a^2))/(x-a) does not exist?
x→a
Where am I wrong in my arguement?
lim f(x)g(x)
x→a
Spivak has proved that this is equal to lim f(x) multlied by
x→a
lim g(x)
x→a
And also if lim g(x) = k and k≠0,
x→a
Then. lim 1/g(x) = 1/k
x→a
Now the problem arises...
Consider the limit
lim ((x^2)-(a^2))/(x-a)
x→a
It can factorised and written as( taking x-2 from numerator)
lim (x+a)
x→a
Which is nothing but 2a.
Now we can write it the above limit also as
lim(x^2)-(a^2) multiplied by
x→a
lim 1/(x-a)
x→a.
The second limit does not exist because
lim(x-a)=0 and l=0
x→a
So, its reciprocal limit does not exist.
Then can't we say
lim ((x^2)-(a^2))/(x-a) does not exist?
x→a
Where am I wrong in my arguement?