Question

evaluate the limit.
lim _ { t
ightarrow infty } left( - 10 t ^ { 2 } - 2 t + 5
ight)
simplify any fractions in your answer.

Explanation:

Step1: Analyze the leading term

For the polynomial \(-10t^{2}-2t + 5\), the leading term is \(-10t^{2}\). As \(t\to\infty\), the behavior of the polynomial is dominated by the leading term.

Step2: Evaluate the limit of the leading term

We know that \(\lim_{t\to\infty}at^{n}=\infty\) if \(a>0\) and \(n>0\), and \(\lim_{t\to\infty}at^{n}=-\infty\) if \(a < 0\) and \(n>0\). Here \(a=-10\) and \(n = 2\). So \(\lim_{t\to\infty}-10t^{2}=-\infty\). Since \(\lim_{t\to\infty}(-10t^{2}-2t + 5)=\lim_{t\to\infty}-10t^{2}+\lim_{t\to\infty}(-2t)+\lim_{t\to\infty}5\), and \(\lim_{t\to\infty}(-2t)=-\infty\), \(\lim_{t\to\infty}5 = 5\). A sum of \(-\infty\) (from \(-10t^{2}\) and \(-2t\)) and a finite number \(5\) is still \(-\infty\).

Answer:

\(-\infty\)