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Chapter 1: Problem 9

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

Expert verified
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

01

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 \(f(x) = \frac{x+2}{2}\).
02

Substitute the target value

Substitute \(x = 3\) into the identified function. Therefore, \(f(3) = \frac{3 + 2}{2}\).
03

Simplify

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

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