Limits

[quaint python]

this limits

of course without the hospital rule and the power series expansion

[frozen bough]

We have:
(x + 1)^(x + 2)/x^(x + 1) = (x + 1)(x + 1)^(x + 1)/x^(x + 1) = (x + 1)((x + 1)/x)^(x +1) = (x + 1)(1 + 1/x)^(x + 1) = (x + 1)(1 + 1/x)(1 + 1/x)^x
So:
-2 + ln((x + 1)^(x + 2)/x^(x + 1)) = -2 + ln(x + 1) + ln(1 + 1/x) + ln((1 + 1/x)^x)
-2 is constant, ln(x + 1) -> +∞, ln(1 + 1/x) -> ln(1) = 0, ln((1 + 1/x)^x) -> ln(e) = 1. So, the limit is +∞.

[quaint python]

thank uuuu so much

[ocean timber]

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