Question

evaluate the limit.

lim_{t→-∞} (-6t² - 3t + 6)/(-6t² - 7t - 3)

simplify any fractions in your answer.

Explanation:

Step1: Divide numerator and denominator by \(t^{2}\)

$$\begin{align*} \lim_{t ightarrow-\infty}\frac{-6t^{2}-3t + 6}{-6t^{2}-7t - 3}&=\lim_{t ightarrow-\infty}\frac{\frac{-6t^{2}}{t^{2}}-\frac{3t}{t^{2}}+\frac{6}{t^{2}}}{\frac{-6t^{2}}{t^{2}}-\frac{7t}{t^{2}}-\frac{3}{t^{2}}}\\ &=\lim_{t ightarrow-\infty}\frac{-6-\frac{3}{t}+\frac{6}{t^{2}}}{-6-\frac{7}{t}-\frac{3}{t^{2}}} \end{align*}$$

Step2: Use the limit property \(\lim_{t

ightarrow\pm\infty}\frac{1}{t^{n}} = 0\) (\(n>0\))
As \(t
ightarrow-\infty\), \(\lim_{t
ightarrow-\infty}\frac{1}{t}=0\) and \(\lim_{t
ightarrow-\infty}\frac{1}{t^{2}} = 0\)

Substitute these values into the fraction:

\(\frac{-6-0 + 0}{-6-0-0}\)

Answer:

\(1\)