ercagpince said:Could you show it explicitly?
ercagpince said:if you take the stirling's approximation there is no need to take log of the term on numerator.
that why i asked that.
The limit of an expression represents the value that the expression approaches as the independent variable approaches a certain value. It is a way to describe the behavior of a function at a specific point or as the input approaches a certain value.
To prove the limit of an expression, you need to show that for any small positive number, ε, there exists a corresponding positive number, δ, such that when the independent variable is within a distance of δ from the limiting value, the expression is within a distance of ε from the limiting value.
The expression n^n/n! is significant because it represents the ratio of the growth rate of an exponential function (n^n) to the growth rate of a factorial function (n!). This ratio is important in understanding the behavior of many mathematical and scientific phenomena.
No, the limit of an expression cannot be proven using algebraic manipulation alone. Algebraic manipulation can help in simplifying an expression and identifying its limiting value, but to prove the limit, we need to use the formal definition of a limit and show that it holds for all values of the independent variable.
The value of n affects the limit of the expression n^n/n! because as n increases, the ratio of n^n/n! also increases. This means that the expression approaches infinity as n increases, and the limiting value will be infinity. Similarly, as n decreases, the expression approaches 0, and the limiting value will be 0.