Question

Find the limit or show that it does not exist.

17. \(\mathop {lim}\limits_{t \to \infty } \frac{{3{t^2} + t}}{{{t^3} - 4t + 1}}\)

Short answer

The value of the limit is \(\infty \).

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

\(\begin{aligned}\mathop {\lim }\limits_{x \to \infty } \frac{{3{t^2} + t}}{{{t^3} - 4t + 1}}{\rm{ }} &= \frac{{3{{\left( \infty \right)}^2} + \infty }}{{{{\left( \infty \right)}^3} - 4\left( \infty \right) + 1}}\\ &= \frac{\infty }{\infty }\\ &= \infty \end{aligned}\)

Thus, the value of the limit is \(\infty \).