To find the limit of the function [tex]\(\lim _{x \rightarrow \infty} \frac{(3 x-1)^8 \cdot(4 x-1)^2}{(3 x+1)^{10}}\)[/tex], we can analyze and simplify the expression as [tex]\( x \)[/tex] approaches infinity. Here's a step-by-step breakdown of the process:
1. Identify Leading Terms:
As [tex]\( x \)[/tex] approaches infinity, the terms involving [tex]\( x \)[/tex] will dominate the constants. Therefore, let's rewrite the terms focusing on their leading coefficients:
- [tex]\((3x - 1)\)[/tex] can be approximated by [tex]\( 3x \)[/tex]
- [tex]\((4x - 1)\)[/tex] can be approximated by [tex]\( 4x \)[/tex]
- [tex]\((3x + 1)\)[/tex] can be approximated by [tex]\( 3x \)[/tex]
2. Rewrite the Expression:
Substitute the leading terms back into the original function:
[tex]\[
\frac{(3 x-1)^8 \cdot (4 x-1)^2}{(3 x +1)^{10}} \approx \frac{(3x)^8 \cdot (4x)^2}{(3x)^{10}}
\][/tex]
3. Simplify the Expression:
Combine the terms in the numerator and denominator:
[tex]\[
\frac{(3x)^8 \cdot (4x)^2}{(3x)^{10}} = \frac{3^8 \cdot x^8 \cdot 4^2 \cdot x^2}{3^{10} \cdot x^{10}}
\][/tex]
4. Factor Out Common Terms:
Combine the exponents of [tex]\( x \)[/tex] and simplify the coefficients:
[tex]\[
\frac{3^8 \cdot 4^2 \cdot x^{8+2}}{3^{10} \cdot x^{10}} = \frac{3^8 \cdot 16 \cdot x^{10}}{3^{10} \cdot x^{10}}
\][/tex]
5. Cancel the Common Terms:
The [tex]\( x^{10} \)[/tex] terms cancel out:
[tex]\[
\frac{3^8 \cdot 16}{3^{10}} = \frac{16}{3^{10-8}} = \frac{16}{3^2}
\][/tex]
6. Final Calculation:
Compute the remaining fraction:
[tex]\[
\frac{16}{3^2} = \frac{16}{9}
\][/tex]
Therefore, the limit is:
[tex]\[
\lim _{x \rightarrow \infty} \frac{(3 x-1)^8 \cdot(4 x-1)^2}{(3 x+1)^{10}} = \frac{16}{9}
\][/tex]
