No. Don't multiply the two factors. Each one has a limit that is readily obtainable. As n gets large without bound, what happens to 1/n2? What happens to 1 + 1/n2? What happens to 7/n? What happens to 7/n + 1?naspek said:Homework Statement
lim... (1 + 1/n^2)(7/n+1)
n->infinity
The Attempt at a Solution
lim... (1 + 1/n^2)(7/n+1)
n->infinity
= (7/n + 1) + (7/n^3 + n^2)
bring out 7 because constant..
7lim... (1/n + 1) + (1/n^3 + n^2)
n->infinity
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7lim... (n^2 + n + 1) / (n^3 + 2n^2 +n)
n->infinity
limit does not exist.. am i correct?
The purpose of evaluating this limit is to determine the behavior of the function as the input (n) approaches a certain value. In this case, we are interested in the behavior of the function as n approaches infinity.
The general method for evaluating a limit is to first try plugging in the value that the input is approaching. If this results in a definite value, then that is the limit. If not, then we need to use algebraic manipulation, factoring, or other techniques to simplify the expression and try again.
To simplify the given expression, we can factor out a common term from both the numerator and denominator. In this case, we can factor out n from the numerator and n^2 from the denominator, giving us (n + 1) / (n^2 + 2n + 1). Then, we can use the fact that (a + b)^2 = a^2 + 2ab + b^2 to rewrite the denominator as (n + 1)^2. This gives us a simpler expression of (n + 1) / (n + 1)^2.
The limit of this simplified expression is 1. As n approaches infinity, both the numerator and denominator approach infinity, but the denominator has a higher degree term. This means that the expression as a whole will approach 0, and the limit will be 1.
The overall limit of the original expression is also 1. Since we simplified the expression to (n + 1) / (n + 1)^2, the behavior of the original expression will be the same as the simplified one. Therefore, both limits will be 1.