Alright, let's explore the limit:
[tex]\[
\lim_{x \to \infty} \frac{(3-1)^8 - (4x - y)^2}{(3x + 1)^{10}}
\][/tex]
Let's approach this step-by-step:
1. Simplify the constant terms in the numerator:
[tex]\[
(3 - 1)^8 = 2^8 = 256
\][/tex]
Therefore, the numerator becomes:
[tex]\[
256 - (4x - y)^2
\][/tex]
2. Expand and simplify the expression inside the numerator:
For very large values of [tex]\( x \)[/tex], the term [tex]\( y \)[/tex] becomes negligible in [tex]\( (4x - y)^2 \)[/tex]. Thus, [tex]\( (4x - y)^2 \)[/tex] approximates to [tex]\( (4x)^2 \)[/tex]:
\]
[tex]\[
(4x)^2 = 16x^2
\][/tex]
Hence, the numerator becomes:
[tex]\[
256 - 16x^2
\][/tex]
3. Simplify the denominator:
[tex]\[
(3x + 1)^{10}
\][/tex]
For very large values of [tex]\( x \)[/tex], the term [tex]\( 1 \)[/tex] becomes negligible, making:
\]
[tex]\(
(3x + 1)^{10} \approx (3x)^{10}
\]
Simplifying further, we get:
\]
(3x)^{10} = 3^{10} \cdot x^{10} = 59049 \cdot x^{10}
\]
4. Rewrite the limit expression with the simplified numerator and denominator:
\]
\(
\lim_{x \to \infty} \frac{256 - 16x^2}{59049 \cdot x^{10}}
\]
5. Simplify the expression by dividing both the numerator and the denominator by \( x^2 \)[/tex]:
\]
[tex]\(
\frac{256 - 16x^2}{59049 \cdot x^{10}} = \frac{256/x^2 - 16x^2/x^2}{59049 \cdot x^{10}/x^2} = \frac{256/x^2 - 16}{59049 \cdot x^8}
\]
6. Take the limit as \( x \)[/tex] approaches infinity:
\]
[tex]\(
\lim_{x \to \infty} \frac{256/x^2 - 16}{59049 \cdot x^8}
\]
As \( x \to \infty \)[/tex], [tex]\(256/x^2\)[/tex] approaches 0, and the expression simplifies to:
\]
[tex]\(
\frac{0 - 16}{59049 \cdot x^8} \rightarrow \frac{-16}{59049 \cdot x^8}
\]
Since \(x^8\)[/tex] in the denominator grows very large, the entire fraction tends towards 0.
Therefore, we conclude:
[tex]\[
\lim_{x \rightarrow \infty} \frac{(3-1)^8-(4 x-y)^2}{(3 x+1)^{10}} = 0
\][/tex]
