Question

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 3^{-}} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} \frac{x+2}{2}, & x \leq 3 \\ \frac{12-2 x}{3}, & x>3 \end{array}\right. $$

Short answer

The limit of the function as \(x\) approaches 3 from the left is 2.5. This means that as we take values of \(x\) that are slightly less than 3, the value of \(f(x)\) gets very close to 2.5.

Step-by-step solution

Identify the relevant function

Since we are trying to evaluate the limit as \(x\) approaches 3 from the left (denoted as \(x \rightarrow 3^{-}\)), we need to use the function definition that applies to values of \(x\) that are less than or equal to 3. In this case, the relevant function is \(f(x) = \frac{x+2}{2}\).

Substitute the target value

Substitute \(x = 3\) into the identified function. Therefore, \(f(3) = \frac{3 + 2}{2}\).

Simplify

Simplify the expression acquired from the substitution. The result is \(f(3) = \frac{5}{2}\), or 2.5 when expressed as a decimal.