Question

Find the limit or show that it does not exist.

30. \(\mathop {lim}\limits_{x \to - \infty } \left( {\sqrt {4{x^2} + 3x} + 2x} \right)\)

Short answer

The limit does not exist.

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:

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

Thus, the value of the limit does not exist.