Carla Murphy

2021-12-27

Evaluating $\underset{n\to \mathrm{\infty}}{lim}{(1+\frac{\mathrm{sin}n}{5n+1})}^{2n+3}$ (a $1}^{\mathrm{\infty}$ indeterminate form)

karton

Expert2022-01-08Added 613 answers

Take logarithm and write it as

the limit becomes

which does not exist.

Observe that

So, your limit does not exist.

user_27qwe

Skilled2022-01-08Added 375 answers

Define

and take the logarithm of both sides to bring the exponent down. Proceed from there.

Vasquez

Expert2022-01-08Added 669 answers

Using

For any n such that

For any n such that

And thus the limit can't exist, as the value is infinitely often less than 0.9 and infinitely often greater than 1.2.

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