Question

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

4. \(\mathop {lim}\limits_{x \to - 3} \left( {2{x^3} + 6{x^2} - 9} \right)\)

Short answer

The limit value is \(\mathop {lim}\limits_{x \to - 3} \left( {2{x^3} + 6{x^2} - 9} \right) = - 9\).

Step-by-step solution

Apply Difference and sum laws

According the Difference and sum law, \(\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right) - g\left( x \right)} \right) = \mathop {\lim }\limits_{x \to a} f\left( x \right) - \mathop {\lim }\limits_{x \to a} g\left( x \right)\) and \(\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right) + g\left( x \right)} \right) = \mathop {\lim }\limits_{x \to a} f\left( x \right) + \mathop {\lim }\limits_{x \to a} g\left( x \right)\), where \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\) and \(\mathop {\lim }\limits_{x \to a} g\left( x \right)\) exists.

Apply the law in given function as:

\(\mathop {lim}\limits_{x \to - 3} \left( {2{x^3} + 6{x^2} - 9} \right) = \mathop {lim}\limits_{x \to - 3} 2{x^3} + \mathop {lim}\limits_{x \to - 3} 6{x^2} - \mathop {lim}\limits_{x \to - 3} 9\)

 Step 2: Apply Constant Multiple laws

According the constant multiple law, \(\mathop {\lim }\limits_{x \to a} cf\left( x \right) = c\mathop {\lim }\limits_{x \to a} f\left( x \right)\), where \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\) exists.

Apply the law as:

\(\mathop {lim}\limits_{x \to - 3} 2{x^3} + \mathop {lim}\limits_{x \to - 3} 6{x^2} - \mathop {lim}\limits_{x \to - 3} 9 = 2\mathop {lim}\limits_{x \to - 3} {x^3} + 6\mathop {lim}\limits_{x \to - 3} {x^2} - \mathop {lim}\limits_{x \to - 3} 9\)

 Step 3: Apply limit laws

According to the law, \(\mathop {\lim }\limits_{x \to a} c = c\), and \(\mathop {\lim }\limits_{x \to a} {x^n} = {a^n}\)where \(n\) is positive integer.

Apply the law, and the values as shown below:

\(\begin{array}{c}2\mathop {lim}\limits_{x \to - 3} {x^3} + 6\mathop {lim}\limits_{x \to - 3} {x^2} - \mathop {lim}\limits_{x \to - 3} 9 &=& 2{\left( { - 3} \right)^3} + 6{\left( { - 3} \right)^2} - 9\\ &=& 2\left( { - 27} \right) + 6\left( 9 \right) - 9\\ &=& - 54 + 54 - 9\\ &=& - 9\end{array}\)

Thus, the value of the limit is \( - 9\).