Limit And Continuity

To ensure is continuous at , the left-hand limit, right-hand limit, and the value of the function at must be equal. Let's calculate these limits:
Left-hand limit as :The function is given by .
As , and .
Performing a Taylor expansion of around :
where .
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Thus, the left-hand limit is .
Right-hand limit as :
The function is given by .
As , .
Using Taylor expansion: .
Let , then ,
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Thus, the right-hand limit is .
For continuity at , left-hand limit = right-hand limit = .
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This implies and .
Finally, .