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Question

Check the continuity of

f(x)={(1+3x)1/xx0e3x=0


at x=0.

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Solution

For f(x) to be continuous at x=0, its limit must be equal to e3 at x=0 or 0+.
limx0(1+3x)1/x=ek......[1 form]
where,
k=limx0(1/x)(1+3x1)=limx03=3
Hence,
limx0(1+3x)1/x=ek=e3
So, function is continuous at x=0.

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