Questions: Evaluate the following limit. lim as t approaches infinity of (t^6 + 2t - 8) / (5t^6 + 3t^3) (A) 5 (B) 1 (C) 0 (D) 1/5 (E) infinity

Solution Steps

To evaluate the limit as \( t \) approaches infinity, we focus on the highest degree terms in the numerator and the denominator. The highest degree term in both the numerator and the denominator is \( t^6 \). By dividing every term by \( t^6 \), we simplify the expression and find the limit.

We need to evaluate the limit: \[ \lim_{t \rightarrow \infty} \frac{t^{6} + 2t - 8}{5t^{6} + 3t^{3}}. \]

To simplify the expression, we divide every term in the numerator and the denominator by \( t^{6} \): \[ \frac{t^{6}/t^{6} + 2t/t^{6} - 8/t^{6}}{5t^{6}/t^{6} + 3t^{3}/t^{6}} = \frac{1 + \frac{2}{t^{5}} - \frac{8}{t^{6}}}{5 + \frac{3}{t^{3}}}. \]

As \( t \) approaches infinity, the terms \( \frac{2}{t^{5}} \), \( \frac{8}{t^{6}} \), and \( \frac{3}{t^{3}} \) all approach \( 0 \). Thus, the limit simplifies to: \[ \frac{1 + 0 - 0}{5 + 0} = \frac{1}{5}. \]

Final Answer