Combination of Logarithm, Limit, and Integral

Level 2

Evaluate $\ln \left|\lim_{n \to \infty} n \left(\int_1^e (1-\sqrt[n]{x})\,dx \right) \right|.$ Note: $\ln$ is the natural logarithm.

The answer is 0.

1 solution

Vivek Bhagat
Mar 18, 2014

This question can be easily solved by L'hospitals rule, all you need to do is that take that n on denominator as 1/n, so here is the case of infinity/infinity, and perfect chance to apply L'hospitals rule, you have to differentiate the integral, (see DUIS) which cancels the (-1/n^2)term from numerator and denominator, then all left is to integrate log(x)*x^(1/n), but as n tends to infinity, x^(1/n)tends to 1. so integrate only logx, which will finally give you the ans to be 0

Combination of Logarithm, Limit, and Integral

The answer is 0.

1 solution

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