Problem 90 - Single Variable Calculus

Question

Let $$ f(x)=\left\\{\begin{array}{ll} |x| & \text { if } x \leq 1 \\ {[x]} & \text { if } x>1 \end{array}\right. $$ Does \(\lim _{x \rightarrow 1} f(x)\) exist? If so, what is its value?

Step-by-Step Solution

Answer

The limit \(\lim _{x \rightarrow 1} f(x)\) exists and its value is 1, as both the left and right limits of the function \(f(x)\) are equal to 1 when approaching 1.

Step 1: Determine the function's value at x = 1

Since the function is defined as \(|x|\) when \(x \leq 1\), we can calculate the value of the function at x = 1. Since the absolute value of 1 is 1, we get \(f(1) = |1| = 1\).

Step 2: Evaluate the limit from the left

As we approach 1 from the left side (values less than 1), we need to consider the first part of the function, \(|x|\). Since the absolute value function is continuous, we can simply find the absolute value of 1: \[\lim _{x \rightarrow 1^{-}} |x| = |1| = 1 \]

Step 3: Evaluate the limit from the right

As we approach 1 from the right side (values greater than 1), we need to use the second part of the function, \([x]\). Since the function \([x]\) simply indicates the integer part of the number (the largest integer not exceeding x), we see that the function remains at 1 for all values greater than 1 but less than 2. Therefore, the limit from the right is: \[\lim _{x \rightarrow 1^{+}} [x] = 1 \]

Step 4: Compare the limits from the left and right

Since both limits from the left and the right are equal, we can now determine the limit of the entire function as \(x\) approaches 1: \[\lim _{x \rightarrow 1} f(x) = 1\] Therefore, the limit exists and its value is 1.

Key Concepts

Absolute Value Function

An absolute value function gives the distance of a number from zero on a number line, irrespective of direction. This function is symbolized by the absolute value symbol \( |x| \). The absolute value of any real number \( x \) is defined as follows:

• \( |x| = x \), if \( x \geq 0 \)
• \( |x| = -x \), if \( x < 0 \)

This means if \( x \) is a positive number or zero, the absolute value remains the same. However, if \( x \) is negative, the sign is flipped to make it positive. The absolute value function is continuous and smooth, which makes it predictable. When assessing limits, particularly as \( x \) approaches a certain point, this predictability helps in determining the behavior of the function. For example, as \( x \) approaches 1 from the left in our original exercise, the limit of \( |x| \) is calculated to be 1, because the value plugged directly into the function remains stable.

Piecewise Function

A piecewise function is a function constructed from multiple distinct functions, each applying to a certain interval of the main function's domain. In our exercise, the function is defined by two expressions that operate over different parts of the domain:

• \( |x| \) when \( x \leq 1 \)
• \( [x] \) when \( x > 1 \)

This setup allows for flexibility in defining a function that behaves differently depending on the input value. Understanding piecewise functions involves:

• Identifying the different expressions correctly and their corresponding intervals.
• Evaluating limits may require separate consideration of each piece over its respective interval.
• Ensuring continuity by checking the values at the boundaries of these intervals.

In the case of our example, we had to look at each portion of the piecewise function separately when approaching \( x = 1 \). The primary challenge with piecewise functions is determining how smoothly or abruptly they transition from one part to another, as seen when examining the limit from both sides.

Step Function

A step function, like the greatest integer function \([x]\), is a type of piecewise function where the output remains constant within certain intervals, abruptly changing to a different value at the next interval. The greatest integer function returns the largest integer less than or equal to \( x \).Consider these key aspects of step functions:

• They produce outputs that are "steps" or flat segments at certain ranges on the graph.
• They are discontinuous at integer points because of sudden changes in function value.
• Handling limits with step functions involves determining the function's behavior on both sides of these discontinuous jumps.

In our exercise, as \( x \) approaches 1 from the right, the function \([x]\) retains the integer value of 1 until \( x \) becomes 2, demonstrating how step functions maintain constant values before stepping to the next level. As a result, when evaluating the limit from the right side, it simplifies to \( 1 \) based on this defined behavior of the greatest integer function.