6.119. lim _(xarrow infty )((7x^10-3)/(7x^10)+2)^-2x^(10)

To solve this limit, we can use the fact that as \( x \) approaches infinity, the dominant term in the numerator and denominator will be the highest power of \( x \). In this case, the dominant term in both the numerator and denominator is \( 7x^{10} \).<br /><br />We can rewrite the expression as:<br /><br />\[<br />\lim_{x \to \infty} \left( \frac{7x^{10} - 3}{7x^{10} + 2} \right)^{-2x^{10}}<br />\]<br /><br />First, let's simplify the fraction inside the limit:<br /><br />\[<br />\frac{7x^{10} - 3}{7x^{10} + 2} \approx \frac{7x^{10}}{7x^{10}} = 1<br />\]<br /><br />However, we need to be more precise. Let's factor out \( 7x^{10} \) from the numerator and denominator:<br /><br />\[<br />\frac{7x^{10} - 3}{7x^{10} + 2} = \frac{7x^{10}(1 - \frac{3}{7x^{10}})}{7x^{10}(1 + \frac{2}{7x^{10}})} = \frac{1 - \frac{3}{7x^{10}}}{1 + \frac{2}{7x^{10}}}<br />\]<br /><br />As \( x \to \infty \), \( \frac{3}{7x^{10}} \) and \( \frac{2}{7x^{10}} \) approach 0, so:<br /><br />\[<br />\frac{1 - \frac{3}{7x^{10}}}{1 + \frac{2}{7x^{10}}} \approx \frac{1 - 0}{1 + 0} = 1<br />\]<br /><br />Thus, the limit simplifies to:<br /><br />\[<br />\lim_{x \to \infty} \left( \frac{7x^{10} - 3}{7x^{10} + 2} \right)^{-2x^{10}} = \left( 1 \right)^{-2x^{10}}<br />\]<br /><br />Since \( 1^{-2x^{10}} = 1 \):<br /><br />\[<br />\lim_{x \to \infty} \left( \frac{7x^{10} - 3}{7x^{10} + 2} \right)^{-2x^{10}} = 1<br />\]<br /><br />Therefore, the correct answer is:<br /><br />\[<br />\boxed{1}<br />\]