how do you take a limit of [ln(1+e^n)]/2n as n goes to infinity?

Take the limit of the ratio of the derivatives of numerator and denominator, with respect to n.

That ratio is the limit of
e^n/[2(1 + e^n)]

which approaches 1/2 since 1 becomes negligible compared to e^n, as n goes to infinity.

To take the limit of the given expression as n goes to infinity, you can use some properties of limits and the properties of the natural logarithm function. Here's the step-by-step explanation:

1. Start by simplifying the expression: [ln(1+e^n)]/2n.
2. Apply the property of the natural logarithm: ln(1+e^n) = n * ln(e) + ln(1) = n + ln(1) = n.
3. Now the expression becomes n/2n.
4. Simplify further by canceling out the common factor of n: 1/2.
5. As n approaches infinity, the limit of 1/2 is 1/2.

So, the limit of [ln(1+e^n)]/2n as n goes to infinity is 1/2.