What is lim_{n→∞} a_n for a_n = n/(n+1)?

What is lim_{n→∞} a_n for a_n = n/(n+1)?

• A. 1
• B. 0
• C. 1/2
• D. undefined

Answer: (A)

The limit of a sequence as n grows without bound is about what value the terms settle toward. For a_n = n/(n+1), the extra 1 in the denominator becomes insignificant as n gets large, so the fraction gets closer to 1. One way to see this is to divide top and bottom by n: a_n = n/(n+1) = 1/(1 + 1/n). Since 1/n → 0, this gives a_n → 1/(1+0) = 1. Another equivalent form is a_n = 1 − 1/(n+1), and since 1/(n+1) → 0, we also get a_n → 1. Thus the limit is 1. The option that equals 1 is the correct choice. The other possibilities, 0, 1/2, or undefined, do not match the actual behavior because the terms do not approach those values and the limit exists.

The limit of a sequence as n grows without bound is about what value the terms settle toward. For a_n = n/(n+1), the extra 1 in the denominator becomes insignificant as n gets large, so the fraction gets closer to 1.

One way to see this is to divide top and bottom by n: a_n = n/(n+1) = 1/(1 + 1/n). Since 1/n → 0, this gives a_n → 1/(1+0) = 1. Another equivalent form is a_n = 1 − 1/(n+1), and since 1/(n+1) → 0, we also get a_n → 1.

Thus the limit is 1. The option that equals 1 is the correct choice. The other possibilities, 0, 1/2, or undefined, do not match the actual behavior because the terms do not approach those values and the limit exists.