what is the limit of (9x/(9x+5))^(6x) as x approaches infinity?
To find the limit of the given expression as x approaches infinity, we can use some algebraic manipulation and the properties of limits. Here's how we can do it step by step:
First, let's rewrite the expression in its exponential form:
(9x / (9x + 5))^(6x) = [(9x + 5 - 5x) / (9x + 5)]^(6x)
= [(9x + 5) / (9x + 5) - (5x / (9x + 5))]^(6x)
Now, let's look at each term separately.
Term 1: (9x + 5) / (9x + 5)
As x approaches infinity, the term [(9x + 5) / (9x + 5)] approaches 1. This is because the numerator and denominator both contain the largest power of x (9x), so all other terms become negligible compared to it.
Term 2: (5x / (9x + 5))
As x approaches infinity, the term (5x / (9x + 5)) approaches 5/9. This is because, as x gets larger and larger, the term 5x grows much faster than the constant term (9x + 5), making the constant term less significant.
Now, let's substitute the limits we found back into our original expression:
[(9x + 5) / (9x + 5) - (5x / (9x + 5))]^(6x) =
[1 - (5/9)]^(6x)
Simplifying further:
= (4/9)^(6x)
Finally, as x approaches infinity, the base (4/9) remains constant, and the exponent 6x increases without bound. Therefore, the limit of the expression is either infinity (if 6x is positive) or zero (if 6x is negative).
In this case, since the value of x is not specified, we cannot determine the exact value of the limit. However, we can conclude that the limit is either infinity or zero, depending on the value of x.