Dick said:You've got this all twisted around. You want 1/sqrt(n)<E for n>N. So you want N>1/E^2.
The limit of x_n is 0 as n approaches infinity. This means that as n gets larger and larger, the value of x_n gets closer and closer to 0.
To prove this limit, we can use the epsilon-delta definition of a limit. This involves showing that for any small positive number ε, we can find a corresponding value of n (denoted as N) such that for all n > N, the absolute value of x_n - 0 is less than ε. This shows that as n gets larger, x_n gets closer and closer to 0.
The value of x_n gets closer to 0 as n approaches infinity because the denominator, which is sqrt(n), gets larger and larger. This means that the fraction 1/sqrt(n) gets smaller and smaller, approaching 0 as n approaches infinity.
Yes, a graph can be used to visually demonstrate the limit of x_n as n approaches infinity. The graph will show that as n gets larger, the value of x_n approaches 0 on the y-axis.
Yes, this limit can be applied in various fields such as physics, engineering, and economics. For example, in physics, this limit can be used to calculate the acceleration of an object as it approaches a constant velocity. In economics, it can be used to analyze the diminishing returns of resources as production increases.