Question

Find the limit or show that it does not exist.

18. \(\mathop {\lim }\limits_{x \to - \infty } \frac{{6{t^2} + t - 5}}{{9 - {t^2}}}\)

Short answer

The value of the limit is \( - 3\).

Step-by-step solution

Use the property of limit

A function is continuous at a point “a” if \(\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\)

Evaluate the given limit

Evaluate the given limit as follows:

\(\begin{aligned}\mathop {\lim }\limits_{x \to - \infty } \frac{{6{t^2} + t - 5}}{{9 - 2{t^2}}}{\rm{ }} &= \mathop {\lim }\limits_{x \to \infty } \frac{{\left( {6 + \frac{1}{t} - \frac{5}{{{t^2}}}} \right)}}{{{t^2}\left( {\frac{9}{{{t^2}}} - 2} \right)}}\\ &= \mathop {\lim }\limits_{x \to \infty } \frac{{\left( {6 + \frac{1}{t} - \frac{5}{{{t^2}}}} \right)}}{{\left( {\frac{9}{{{t^2}}} - 2} \right)}}\\ &= \frac{{6 + \frac{1}{\infty } - \frac{5}{{{t^2}}}}}{{\frac{9}{{{\infty ^2}}} - 2}}\\ &= \frac{{6 + 0 - 0}}{{0 - 2}}\\ &= \frac{6}{{ - 2}}\\ &= - 3\end{aligned}\)

Thus, the value of the limit is \( - 3\).