Questions: Question 1 (4 points) Evaluate the following limit: lim x -> -∞ (-1-3x+4x^2) / (-6+11x-3x^2) A) -1 B) 1/6 C) -4/3 D) -∞ E) 0

Solution Steps

To evaluate the limit as \( x \) approaches \(-\infty\), we should first identify the highest degree terms in the numerator and the denominator, as these will dominate the behavior of the function. In this case, both the numerator and the denominator are quadratic polynomials. We can simplify the expression by dividing every term by \( x^2 \), the highest power of \( x \) in the expression. This will allow us to evaluate the limit by focusing on the leading coefficients of the highest degree terms.

We need to evaluate the limit: \[ \lim_{x \rightarrow -\infty} \frac{-1 - 3x + 4x^2}{-6 + 11x - 3x^2} \]

The highest degree terms in both the numerator and the denominator are \(4x^2\) and \(-3x^2\), respectively. We can simplify the expression by dividing every term by \(x^2\): \[ \frac{\frac{-1}{x^2} - \frac{3}{x} + 4}{-\frac{6}{x^2} + \frac{11}{x} - 3} \]

As \(x\) approaches \(-\infty\), the terms \(\frac{-1}{x^2}\), \(\frac{3}{x}\), \(\frac{6}{x^2}\), and \(\frac{11}{x}\) all approach \(0\). Thus, the limit simplifies to: \[ \lim_{x \rightarrow -\infty} \frac{4}{-3} = \frac{-4}{3} \]

Final Answer