- #1
KTiaam
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Homework Statement
What is the limit of the given equations as n approaches infinity?
(1 + 3n-1)/3n
KTiaam said:Homework Statement
What is the limit of the given equations as n approaches infinity?
(1 + 3n-1)/3n
Dick said:You are supposed to show an attempt to solve it. Try it. Break it into two fractions.
KTiaam said:lim 1/3n + lim 3n-1/3n
n→∞ n→∞
the second equation:
you can cancel the 3n-1
and the second equations turns out to be the same as the first so that equals:
lim 1/3n
n→∞
so in approaches e?
Dick said:e? No, what does 1/3^n approach? Just think about it. And 3^(n-1)/3^n isn't the same as the first term. What is it?
KTiaam said:1/3n approaches 1/ a really big number.
1 over a big number equals 0.
(3n-1)/(3n)
thinking about this equation and plugging in values like 2, 3, 4, 5, ect.
you get
31/32
32/33
33/34
they all equal 1/3
so 1/3?
Dick said:Yes. But you didn't need to plug numbers in. 3^n=3^(n-1)*3. So 3^(n-1)/3^n=1/3.
HallsofIvy said:Where did you get that "0" you are multiplying?
KTiaam said:so it the answer is zero?
1/3 x 0
The limit as n approaches infinity is a mathematical concept that represents the value that a function or sequence approaches as the input variable approaches infinity. It is denoted as limn→∞ f(n) and is used to describe the behavior of a function or sequence as the input variable becomes infinitely large.
The limit as n approaches infinity can be calculated by evaluating the function or sequence at increasingly larger values of n. If the values of the function or sequence approach a single value as n becomes larger and larger, then that value is the limit as n approaches infinity.
The limit as n approaches infinity is important in understanding the behavior of a function or sequence as the input variable becomes extremely large. It helps to determine the existence and value of a horizontal asymptote, which is a line that the function approaches but does not intersect as the input variable approaches infinity.
Yes, the limit as n approaches infinity can be negative, positive, or zero. The sign of the limit depends on the behavior of the function or sequence as the input variable becomes infinitely large. If the values of the function or sequence approach a negative number, then the limit will also be negative.
The concept of limit as n approaches infinity is used in various fields like physics, economics, and engineering to model and predict the behavior of systems as they become increasingly large. It is also used in calculus to solve problems related to rates of change and optimization. For example, the concept of limit as n approaches infinity is used in the study of radioactive decay to determine the half-life of a substance.