Question

Evalauate the limit, if it exists.

\(\mathop {{\bf{lim}}}\limits_{x \to - {\bf{2}}} \left( {{\bf{3}}x - {\bf{7}}} \right)\)

Short answer

The value of the limit is \( - 13\).

Step-by-step solution

Step 1:The limit laws

For the expression \(\mathop {\lim }\limits_{x \to - 2} \left( {3x - 7} \right)\),Difference law is applicable.

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

Evaluate the limit

Solve the expression \(\mathop {\lim }\limits_{x \to - 2} \left( {3x - 7} \right)\).

\(\begin{aligned}\mathop {\lim }\limits_{x \to - 2} \left( {3x - 7} \right) &= \mathop {\lim }\limits_{x \to - 2} \left( {3x} \right) - \mathop {\lim }\limits_{x \to - 2} 7\\ &= 3\left( { - 2} \right) - 7\\ &= - 6 - 7\\ &= - 13\end{aligned}\)

So, the value of the limit is \( - 13\).