Limit Analysis: Finding the Limit of (3^x 5^x 7^x)^(1/x) as x Approaches Infinity

When dealing with limits of complex expressions as x approaches infinity, breaking down the problem can make it much more manageable. In this article, we will explore the limit of the expression (3x 5x 7x)1/x as x tends to infinity, using both intuitive methods and more rigorous approaches like L'H?pital's Rule.

Introduction to the Problem

Let's consider the limit of the expression (3x * 5x * 7x)1/x as x approaches infinity.

Intuitive Approach: Dominant Term Dominance

When x is very large, the term 7x will dominate 3x and 5x. This is because 7 is the largest of the three base numbers, and its exponential growth will overshadow the growth of the others. Therefore, we can approximate the expression as follows:

3x * 5x * 7x ≈ 7x

Thus, the limit can be rewritten and simplified as:

limx→∞ (3x * 5x * 7x)1/x ≈ limx→∞ (7x)1/x

Further simplification yields:

(7x)1/x 7x/x 71 7

Rigorous Approach: Using L'H?pital's Rule

For a more rigorous approach, we can use L'H?pital's Rule. First, let's rewrite the expression using the natural logarithm:

limx→∞ (3x * 5x * 7x)1/x limx→∞ exp{[ln(3x * 5x * 7x)]/x}

Now, we can take the limit of the logarithm of the expression inside the exponent:

limx→∞ [ln(3x * 5x * 7x)]/x limx→∞ (ln(3^x) ln(5^x) ln(7^x))/x

Breaking it down:

limx→∞ (x*ln(3) x*ln(5) x*ln(7))/x

limx→∞ (ln(3) ln(5) ln(7)) ln(3*5*7)

ln(105)

Since the limit of the logarithm is ln(105), applying the exponential function to both sides gives us:

exp(ln(105)) 105

However, we can simplify with the dominant term approach as well, noting that:

limx→∞ (3^x * 5^x * 7^x)^(1/x) exp{limx→∞ [(ln(3^x) ln(5^x) ln(7^x))/x]}

Here, the dominant term is ln(7^x), which simplifies to ln(7).

Conclusion

By both intuitive and rigorous methods, we can conclude that:

limx→∞ (3^x * 5^x * 7^x)^(1/x) 7

The final answer is boxed{7}.