Problem 64

Question

Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ \begin{array}{l} f(x)=\left\\{\begin{array}{ll} 3 x-4, & \text { for } x<1 \\ x-2, & \text { for } x>1 \end{array}\right. \\ \text { Find } \lim _{x \rightarrow 1^{-}} f(x), \lim _{x \rightarrow 1^{+}} f(x), \text { and } \lim _{x \rightarrow 1} f(x) . \end{array} $$

Step-by-Step Solution

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Answer

The limits are $ \lim_{x \to 1^{-}} f(x) = -1 $, $ \lim_{x \to 1^{+}} f(x) = -1 $, and $ \lim_{x \to 1} f(x) = -1 $.

1Step 1: Identify the Piecewise Function

Given the piecewise function,\[ f(x) = \begin{cases} 3x-4, & \text{for } x<1 \ x-2, & \text{for } x>1 \end{cases} \] we see that the function has two different expressions depending on whether $ x $ is less than or greater than 1.

2Step 2: Analyze the Left-Hand Limit at x=1

For $ x \to 1^{-} $, we use the expression for $ x<1 $:\[ f(x) = 3x - 4. \] Therefore,\[ \lim_{x \to 1^{-}} f(x) = \lim_{x \to 1^{-}} (3x - 4) = 3(1) - 4 = -1. \]

3Step 3: Analyze the Right-Hand Limit at x=1

For $ x \to 1^{+} $, we use the expression for $ x>1 $:\[ f(x) = x - 2. \] Therefore,\[ \lim_{x \to 1^{+}} f(x) = \lim_{x \to 1^{+}} (x - 2) = 1 - 2 = -1. \]

4Step 4: Determine the Two-Sided Limit at x=1

Since the left-hand limit ($-1$) and the right-hand limit ($-1$) are equal, the two-sided limit exists and is given by:\[ \lim_{x \to 1} f(x) = -1. \]

5Step 5: Graph the Function

Graph each piece of the function separately. For $ x < 1 $, use the line $ y = 3x - 4 $ which continues leftward. For $ x > 1 $, use the line $ y = x - 2 $ which extends rightward. Note that these lines do not connect at $ x = 1 $, as they are defined only for their respective domains.

Key Concepts

Piecewise FunctionsLeft-Hand LimitRight-Hand LimitTwo-Sided Limit

Piecewise Functions

Piecewise functions are a special type of function used frequently in calculus and algebra. They consist of multiple sub-functions, each with its own domain. This means the function takes different forms depending on the interval of the input value. Consider our example: the function $ f(x) $ is defined differently for $ x < 1 $ and $ x > 1 $.

For $ x < 1 $, the expression is $ 3x - 4 $.
For $ x > 1 $, it shifts to $ x - 2 $.

Each piece of the function behaves like a separate function with a unique equation. This segmentation can appear confusing at first, but breaking it down into sections helps to manage and analyze the behavior of the function across its entire range.

Left-Hand Limit

The left-hand limit focuses on approaching a specific point from the left on the number line. When we write $ \lim_{x \to 1^{-}} f(x) $, we are interested in what value $ f(x) $ approaches as $ x $ gets infinitesimally close to 1 from values less than 1.In our example, for $ x < 1 $, the relevant piece of our piecewise function is $ 3x-4 $.

By substituting 1 into this part of the function, $ 3(1) - 4 = -1 $, we find that the left-hand limit is $-1$.

Understanding this concept is crucial for analyzing the behavior of functions around certain points, particularly in identifying discontinuities or consistent behaviors at those points.

Right-Hand Limit

In contrast to the left-hand limit, the right-hand limit examines the behavior of a function as the input value approaches from the right, values slightly greater than the point of interest.For $ \lim_{x \to 1^{+}} f(x) $, we consider $ x > 1 $. The expression $ x-2 $ becomes relevant here.

Upon substitution, $ 1 - 2 = -1 $, revealing that the right-hand limit is also $-1$.

When both left-hand and right-hand limits are established, they set the stage for determining the two-sided limit, which could further define the overall behavior of the function.

Two-Sided Limit

The two-sided limit examines the overall approach towards a function from both left and right sides of a point. This limit is crucial for concluding if the function behaves consistently around a specific point, especially in determining continuity.In the given piecewise function, we observed that both the left-hand limit at $ x=1 $ and the right-hand limit are $-1$.

Since $ \lim_{x \to 1^{-}} f(x) = -1 $ and $ \lim_{x \to 1^{+}} f(x) = -1 $, the two-sided limit $ \lim_{x \to 1} f(x) $ also equals $-1$.

A consistent two-sided limit indicates that the function does not have a jump or break at that point, thereby confirming the function approaches the same value from both sides.

Problem 64

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