Question

Find the limit or show that it does not exist.

21. \(\mathop {lim}\limits_{t \to \infty } \left( {\sqrt {25{t^2}} + 2 - 5t} \right)\)

Short answer

The value of the limit is \(0\).

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:

\(\begin{aligned}\mathop {\lim }\limits_{x \to \infty } \frac{{3{t^2} + t}}{{{t^3} - 4t + 1}}{\rm{ }} &= \frac{{{t^2}\left( {3 + \frac{1}{t}} \right)}}{{{t^3}\left( {1 - \frac{4}{{{t^2}}} + \frac{1}{{{t^3}}}} \right)}}\\ &= \mathop {\lim }\limits_{x \to \infty } \frac{{\left( {3 + \frac{1}{t}} \right)}}{{t\left( {1 - \frac{4}{\infty } + \frac{1}{{{t^3}}}} \right)}}\\ &= \frac{{3 + 0}}{{\infty \left( {1 - 0 + 0} \right)}}\\ &= \frac{3}{\infty }\\ &= 0\end{aligned}\)

Thus, the value of the limit is \(0\).