Question

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 1^{+}} f(x), \text { where } f(x)=\left\\{\begin{array}{ll}{x,} & {x \leq 1} \\ {1-x,} & {x>1}\end{array}\right. $$

Short answer

The limit of \( f(x) \) as \( x \rightarrow 1^{+} \) is 0.

Step-by-step solution

Identify the Correct Part of the Function

First, identify which part of the function definition is applicable to the limit. In this case, as \( x \) is approaching 1 from the positive direction, the relevant definition is \( f(x) = 1 - x \).

Calculate the Limit

Insert the value into the equation \( f(x) = 1 - x \). This gives us \( f(x) = 1 - 1 = 0 \) when \( x = 1 \).

Confirm the Result

The result is zero because the value of \( f(x) \) tends to become zero as \( x \) tends to 1 from the positive direction.

Key concepts

Limit of a Function

The concept of a limit lies at the heart of calculus, serving as a fundamental building block for further topics such as derivatives and integrals. A limit describes the value that a function approaches as the input (or 'x' value) approaches a certain point. Often expressed as \( \lim_{x \to a} f(x) \), it represents what value the function \((f(x))\) gets closer to as \((x)\) gets infinitesimally close to \((a)\), but not necessarily equal to \((a)\).

Calculating a limit involves understanding the behavior of the function near the point of interest. In some cases, you can plug the target value directly into the function to find the limit. However, not all functions allow for direct substitution, especially if they involve undefined expressions like dividing by zero. In these scenarios, finding limits might require additional techniques such as factoring, rationalizing, or applying special limit laws. Understanding the precise behavior of functions as inputs approach certain values is an essential skill in calculus.

One-Sided Limit

A one-sided limit examines the behavior of a function as the variable \((x)\) approaches a specific value from one side only—either from the left \( (x \to a^{-}) \) or from the right \( (x \to a^{+}) \). These are symbolized as \( \lim_{x \to a^-} f(x) \) and \( \lim_{x \to a^+} f(x) \) respectively. In our exercise, we dealt with the one-sided limit from the right: \( \lim_{x \rightarrow 1^{+}} f(x) \).

The value of a one-sided limit may differ significantly from a two-sided limit, where \((x)\) can approach the point \((a)\) from both sides. Piecewise functions, functions defined by multiple sub-functions over certain intervals, often illustrate different behaviors on each side of a point. This characteristic makes the concept of one-sided limits especially useful. In the provided example, analyzing how the function behaves as \((x)\) approaches 1 from the right is crucial because the function changes its formula at \((x = 1)\).

Piecewise Functions

A piecewise function is a function that is defined by different expressions based on different intervals of the input variable. These functions can resemble a puzzle, with each piece representing a different rule that applies to specific sections of the domain.

For instance, the function given in the exercise is a piecewise function with two pieces: \((f(x) = x)\) for \((x \leq 1)\) and \((f(x) = 1 - x)\) for \((x > 1)\). To find a limit involving a piecewise function, it's essential to first identify which piece of the function you should consider, based on the limit's direction of approach (to the left or the right of the target value). Once the appropriate piece is selected, then you can proceed with finding the limit as you would for any other type of function. Keep in mind that for a limit to exist at a point where a function changes formula, the one-sided limits from both directions must both converge to the same value.