QuestionAugust 25, 2025

Evaluate the limit lim _(xarrow infty )((4-x)(6+5x))/((3-10x)(6+10x)) square

Evaluate the limit lim _(xarrow infty )((4-x)(6+5x))/((3-10x)(6+10x)) square
Evaluate the limit lim _(xarrow infty )((4-x)(6+5x))/((3-10x)(6+10x)) square

Solution
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Answer

\frac{1}{100} Explanation 1. Simplify the expression Expand the numerator and denominator: (4-x)(6+5x) = 20x - x^2 + 30 - 5x and (3-10x)(6+10x) = -100x^2 + 60x - 30. 2. Identify dominant terms The dominant term in the numerator is -x^2, and in the denominator is -100x^2. 3. Calculate the limit Divide both the numerator and the denominator by x^2: \lim_{x \to \infty} \frac{-x^2/x^2}{-100x^2/x^2} = \frac{-1}{-100}.

Explanation

1. Simplify the expression<br /> Expand the numerator and denominator: $(4-x)(6+5x) = 20x - x^2 + 30 - 5x$ and $(3-10x)(6+10x) = -100x^2 + 60x - 30$.<br />2. Identify dominant terms<br /> The dominant term in the numerator is $-x^2$, and in the denominator is $-100x^2$.<br />3. Calculate the limit<br /> Divide both the numerator and the denominator by $x^2$: $\lim_{x \to \infty} \frac{-x^2/x^2}{-100x^2/x^2} = \frac{-1}{-100}$.
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