Question

evaluate the limit
lim_{t \to infty} \frac{7t + 2}{-5t^{2} - 2t - 9}
simplify any fractions in your answer.

Explanation:

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

$$\lim_{t ightarrow\infty}\frac{\frac{7t}{t^{2}}+\frac{2}{t^{2}}}{\frac{-5t^{2}}{t^{2}}-\frac{2t}{t^{2}}-\frac{9}{t^{2}}}=\lim_{t ightarrow\infty}\frac{\frac{7}{t}+\frac{2}{t^{2}}}{- 5-\frac{2}{t}-\frac{9}{t^{2}}}$$

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

ightarrow\infty}\frac{1}{t^{n}} = 0\) (\(n>0\))
As \(t
ightarrow\infty\), \(\lim_{t
ightarrow\infty}\frac{7}{t}=0\), \(\lim_{t
ightarrow\infty}\frac{2}{t^{2}} = 0\), \(\lim_{t
ightarrow\infty}\frac{2}{t}=0\) and \(\lim_{t
ightarrow\infty}\frac{9}{t^{2}}=0\)

Substitute these values into the fraction: \(\frac{0 + 0}{-5-0 - 0}\)

Answer:

\(0\)