Consider the limit(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Spivak problem on limits

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