Problem 54 - College Algebra and Calculus: An Applied Approach

Question

find the limit $$ \lim _{x \rightarrow 1} f(s), \text { where } f(s)=\left\\{\begin{array}{ll} s, & s \leq 1 \\ 1-s, & s>1 \end{array}\right. $$

Step-by-Step Solution

Answer

The limit of the function as x approaches 1 does not exist

Step 1: Evaluate the Left-Hand Limit

Determine the limit of the function as x approaches 1 from the left (using the definition of the function for values less than or equal to 1). This gives: \[ \lim _{x \rightarrow 1^-} f(s) = 1 \]

Step 2: Evaluate the Right-Hand Limit

Next, calculate the limit from the right as x goes to 1 (using the definition for values greater than 1): \[ \lim _{x \rightarrow 1^+} f(s) = 1-1 = 0 \]

Step 3: Compare the Two Limits

Since the left-hand limit (1) and the right-hand limit (0) are different, \[ \lim _{x \rightarrow 1} f(s) \] does not exist.

Key Concepts

Piecewise Function

A piecewise function is a type of function that is defined by different expressions based on the input value. It's like a rulebook that changes depending on what region or interval your input value falls within.
For instance, imagine having a function that tells you how much to pay based on your age. For children, there might be one rate, and for adults, another. That's a piecewise function in action!
In the exercise, the piecewise function is defined as follows:

• If \( s \leq 1 \), the function equals \( s \).
• If \( s > 1 \), it changes to \( 1-s \).

Understanding the structure of a piecewise function is crucial because each piece has its own behavior and rule for calculating limits.
You always need to check each segment separately when analyzing limits or other properties.

Left-Hand Limit

The left-hand limit is a concept used to evaluate how a function behaves as the input approaches a certain value from the left side. It's like sneaking up on a number from smaller numbers, getting closer and closer.
The mathematical notation is \( \, \lim _{x \rightarrow a^-} f(x) \, \), which reads "the limit of \( f(x) \) as \( x \) approaches \( a \) from the left."
In our exercise, we had to figure out what happens to \( f(s) \) as \( s \rightarrow 1 \) from the left. Looking at the function definition:

• For \( s \leq 1 \), the function \( f(s) = s \).

Therefore, as \( s \) approaches 1 from the left, \( f(s) \) approaches \( 1 \). This gives us the left-hand limit to be 1.

Right-Hand Limit

The right-hand limit is another crucial aspect when determining the limit of a function. It tells us how the function behaves as the input value approaches a certain point from the right side, meaning sneaking up on the number from larger numbers.
The notation for this is \( \, \lim _{x \rightarrow a^+} f(x) \, \). This reads as "the limit of \( f(x) \) as \( x \) approaches \( a \) from the right."
In the problem, the function changes its rule when \( s > 1 \), which means:

• For \( s > 1 \), it is defined as \( f(s) = 1-s \).

As \( s \rightarrow 1 \) from the right side, \( f(s) \) approaches \( 0 \). Hence, the right-hand limit of the function is 0.
Recognizing this difference between left and right-hand limits is essential, particularly in showing whether the overall limit exists.