To find the limit [tex]\(\lim_{x \rightarrow \infty} \frac{(2 x+3)^6(3 x-5)^4}{(2 x+1)^{10}}\)[/tex], we should analyze the behavior of the given expression as [tex]\(x\)[/tex] approaches infinity.
1. Identify the dominant terms:
As [tex]\(x\)[/tex] approaches infinity, the dominant terms in the polynomials will be the highest degree terms. For large values of [tex]\(x\)[/tex], the [tex]\(3\)[/tex], [tex]\(-5\)[/tex], and [tex]\(1\)[/tex] in the expressions become negligible compared to the terms involving [tex]\(x\)[/tex]. Therefore, we approximate each part with their highest-degree terms:
[tex]\[
2x + 3 \approx 2x \, \ (as \, \ x \rightarrow \infty)
\][/tex]
[tex]\[
3x - 5 \approx 3x \, \ (as \, \ x \rightarrow \infty)
\][/tex]
[tex]\[
2x + 1 \approx 2x \, \ (as \, \ x \rightarrow \infty)
\][/tex]
2. Simplify the expression under these approximations:
Substituting these approximations into the original expression:
[tex]\[
\frac{(2x+3)^6(3x-5)^4}{(2x+1)^{10}} \approx \frac{(2x)^6(3x)^4}{(2x)^{10}}
\][/tex]
3. Combine and simplify the powers of [tex]\(x\)[/tex]:
Now, let's combine the powers of [tex]\(x\)[/tex]:
[tex]\[
(2x)^6 = 2^6 x^6 = 64 x^6
\][/tex]
[tex]\[
(3x)^4 = 3^4 x^4 = 81 x^4
\][/tex]
[tex]\[
(2x)^{10} = 2^{10} x^{10} = 1024 x^{10}
\][/tex]
Putting these into our simplified fraction:
[tex]\[
\frac{64 x^6 \cdot 81 x^4}{1024 x^{10}} = \frac{5184 x^{10}}{1024 x^{10}}
\][/tex]
4. Cancel out the [tex]\(x^{10}\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{5184 x^{10}}{1024 x^{10}} = \frac{5184}{1024}
\][/tex]
5. Simplify the numerical fraction:
[tex]\[
\frac{5184}{1024} = \frac{81 \times 64}{16 \times 64} = \frac{81}{16}
\][/tex]
Thus, the limit as [tex]\( x \)[/tex] approaches infinity is:
[tex]\[
\lim_{x \rightarrow \infty} \frac{(2 x+3)^6(3 x-5)^4}{(2 x+1)^{10}} = \frac{81}{16}
\][/tex]
