To solve the limit [tex]\(\lim_{x \rightarrow \infty} \frac{(2x + 3)^6 (3x - 5)^4}{(2x + 1)^{10}}\)[/tex], we need to analyze how each part of the expression behaves as [tex]\(x\)[/tex] approaches infinity.
1. Approximate the dominant terms:
- As [tex]\(x\)[/tex] approaches infinity, the linear terms [tex]\(2x\)[/tex], [tex]\(3x\)[/tex], and [tex]\(2x\)[/tex] in the expressions will dominate over the constants 3, -5, and 1, respectively.
- Therefore:
[tex]\[
2x + 3 \approx 2x \quad \text{(since 3 becomes negligible)}
\][/tex]
[tex]\[
3x - 5 \approx 3x \quad \text{(since -5 becomes negligible)}
\][/tex]
[tex]\[
2x + 1 \approx 2x \quad \text{(since 1 becomes negligible)}
\][/tex]
2. Substitute the approximations into the expression:
[tex]\[
\lim_{x \to \infty} \frac{(2x + 3)^6 (3x - 5)^4}{(2x + 1)^{10}} \approx \lim_{x \to \infty} \frac{(2x)^6 (3x)^4}{(2x)^{10}}
\][/tex]
3. Simplify the expression:
- Rewrite the expression to separate the constants and the variable terms:
[tex]\[
\frac{(2x)^6 (3x)^4}{(2x)^{10}} = \frac{2^6 x^6 \cdot 3^4 x^4}{2^{10} x^{10}} = \frac{2^6 \cdot 3^4 \cdot x^{6+4}}{2^{10} \cdot x^{10}}
\][/tex]
- Combine the exponents of [tex]\(x\)[/tex]:
[tex]\[
= \frac{2^6 \cdot 3^4 \cdot x^{10}}{2^{10} \cdot x^{10}}
\][/tex]
- Since [tex]\(x^{10}\)[/tex] appears in both the numerator and the denominator, it cancels out:
[tex]\[
= \frac{2^6 \cdot 3^4}{2^{10}}
\][/tex]
4. Simplify the constants:
- Calculate the powers of the constants:
[tex]\[
2^6 = 64 \quad \text{and} \quad 3^4 = 81
\][/tex]
- So,
[tex]\[
\frac{64 \cdot 81}{2^{10}}
\][/tex]
- Since [tex]\(2^{10} = 1024\)[/tex], the expression becomes:
[tex]\[
\frac{64 \cdot 81}{1024}
\][/tex]
5. Calculate the final value:
- Compute the product and then divide:
[tex]\[
64 \cdot 81 = 5184
\][/tex]
[tex]\[
\frac{5184}{1024} = 5.0625
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{5.0625}
\][/tex]
