Problem 17 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 3^{-}} f(x), \text { where } f(x)=\left\\{\begin{array}{ll}{\frac{x+2}{2},} & {x \leq 3} \\ {\frac{12-2 x}{3},} & {x>3}\end{array}\right. $$

Step-by-Step Solution

Answer

The limit as x approaches 3 from the left, \( \lim _{x \rightarrow 3^{-}} f(x)\), is 2.5.

Step 1: Identify the Relevant Expression

As we are looking for the limit as x approaches 3 from the left (indicated by the supercript '-'), we are only interested in the expression of f(x) valid for x-values less than or equal to 3. This is \(\frac{x+2}{2}\).

Step 2: Substitution into the Expression

Having identified the relevant expression, the next step is to substitute the value 3 into \(\frac{x+2}{2}\) and compute the result.

Step 3: Calculate the Limit

After substituting 3 into \(\frac{x+2}{2}\), we get \(\frac{3+2}{2} = \frac{5}{2} = 2.5\). Therefore, the limit of f(x) as x approaches 3 from the left is 2.5.

Key Concepts

Piecewise Functions

Piecewise functions are a type of mathematical function defined by different expressions, depending on the input value. They are written in parts, each with its own specific condition or interval. For instance, the function mentioned in the exercise is a piecewise function because it has two expressions, each applicable to different domains of x.

• The expression \(\frac{x+2}{2}\) is used when \(x\) is less than or equal to 3.
• The expression \(\frac{12-2x}{3}\) is used when \(x\) is greater than 3.

By defining functions in this way, mathematicians can represent complex scenarios in a simple form. It allows varying outputs for different segments of the input range or domain. Understanding how each piece of the function operates is key to analyzing the function, finding limits, and ensuring the function’s applicability in different contexts.

Continuity

Continuity in calculus roughly describes how smooth a function is. A function is continuous at a point if there is no sudden jump, break, or hole at that point. For a function to be continuous at a particular point, the limit of the function as it approaches that point from either side must equal the function's value at that point. In the context of a piecewise function, continuity is particularly intricate at the boundaries where the expressions change.

• To check continuity at a point \(x = c\), ensure that \(\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)\).

In the given exercise, because we're interested in the left-hand limit of \(f(x)\), we're only concerned with the expression applicable as \(x\) approaches 3 from the subdomain \(x \leq 3\). This allows us to focus only on the left side without testing continuity across both segments.

Left-Hand Limit

The left-hand limit of a function refers to the value that the function approaches as the input variable x approaches a specific point from the left. In symbols, this is written as \(\lim_{x \to a^-} f(x)\). It specifically considers values of \(x\) that are less than \(a\). The importance of identifying the left-hand limit becomes clear when analyzing functions at points where behavior might differ subtly from the left to the right.

• For \(x\) approaching 3 from the left, the relevant expression in the piecewise function is \(\frac{x+2}{2}\).
• Calculating the left-hand limit involves substituting into this expression, which we found gave us a result of 2.5.

The left-hand limit is an essential concept not just in verifying the behavior of piecewise functions, but also in determining continuity and differentiability at given points.