find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the li\n\\( \\lim _ { t…

find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the li\n\\( \\lim _ { t \\rightarrow - \\infty } \\frac { 2 t ^ { 2 } + t } { t ^ { 3 } - 7 t + 1 } \\)

find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the li\n\\( \\lim _ { t \\rightarrow - \\infty } \\frac { 2 t ^ { 2 } + t } { t ^ { 3 } - 7 t + 1 } \\)

Answer

Explanation:

Step1: Divide numerator and denominator by highest power of (t)

Divide numerator (2t^{2}+t) and denominator (t^{3}-7t + 1) by (t^{3}). [ \begin{align*} \frac{2t^{2}+t}{t^{3}-7t + 1}&=\frac{\frac{2t^{2}}{t^{3}}+\frac{t}{t^{3}}}{\frac{t^{3}}{t^{3}}-\frac{7t}{t^{3}}+\frac{1}{t^{3}}}\ &=\frac{\frac{2}{t}+\frac{1}{t^{2}}}{1-\frac{7}{t^{2}}+\frac{1}{t^{3}}} \end{align*} ]

Step2: Apply limit as (t\to-\infty)

We know that (\lim_{t\to-\infty}\frac{1}{t^{n}} = 0) for (n>0). [ \begin{align*} \lim_{t\to-\infty}\frac{\frac{2}{t}+\frac{1}{t^{2}}}{1-\frac{7}{t^{2}}+\frac{1}{t^{3}}}&=\frac{\lim_{t\to-\infty}(\frac{2}{t}+\frac{1}{t^{2}})}{\lim_{t\to-\infty}(1-\frac{7}{t^{2}}+\frac{1}{t^{3}})}\ &=\frac{0 + 0}{1-0 + 0} \end{align*} ]

Answer:

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