lim from n toward infinity
[1+ (-1)^n/n ]
replace n with inf.
1 + (-1^0/0) = 1 +(-1^0) = 1+1= 2
I assume the question is asking
lim from n toward infinity (note additional parentheses):
[1+ ((-1)^n)/n ]
From the basic laws on limites, we can take this as:
Lim [1+((-1)^n/n]
= Lim 1 + Lim (-1)^n*(1/n)
= 1 + Lim (-1)^n * Lim (1/n)
= 1 + Lim (-1)^n * 0
= 1
The Lim (-1)^n is indeterminate. The upper bound is 1 and the lower bound is -1. It does not matter when it is multiplied by zero.
To solve this limit, we can split it into two cases: when n is even and when n is odd.
Case 1: When n is even
For even values of n, (-1)^n will be equal to 1. So, the given expression becomes:
[1 + 1/n]
Taking the limit as n approaches infinity:
lim(n → ∞) [1 + 1/n] = 1 + lim(n → ∞) 1/n = 1 + 0 = 1
Case 2: When n is odd
For odd values of n, (-1)^n will be equal to -1. So, the given expression becomes:
[1 - 1 / n]
Taking the limit as n approaches infinity:
lim(n → ∞) [1 - 1/n] = 1 - lim(n → ∞) 1/n = 1 - 0 = 1
Since the limit is the same in both cases, we can conclude that the overall limit as n approaches infinity is 1.
To determine this, we substituted the value of (-1)^n in the given expression based on whether n is even or odd. Then, we simplified the expression and evaluated the limit as n approaches infinity separately for each case. Finally, we compared the results to find the overall limit.