Question

In Exercises 53 and \(54,\) find the limit of \(f(x)\) as (a) \(x \rightarrow-\infty\) , (b) \(x \rightarrow \infty,(\mathbf{c}) x \rightarrow 0^{-},\) and \((\mathbf{d}) x \rightarrow 0^{+}\) . $$f(x)=\left\\{\begin{array}{ll}{1 / x,} & {x<0} \\ {-1,} & {x \geq 0}\end{array}\right.$$

Short answer

The limit of the function \(f(x)\) as \(x \rightarrow -\infty\) is 0, as \(x \rightarrow \infty\) is -1, as \(x \rightarrow 0^{-}\) is \(\infty\), and as \(x \rightarrow 0^{+}\) is -1.

Step-by-step solution

Finding the limit as \(x\) approaches \(-\infty\)

Since \(x<0\) when \(x\) approaches \(-\infty\), we will use \(1/x\) part. Therefore, the limit as \(x\) approaches \(-\infty\) for the function \(f(x) = 1 / x\) is 0.

Finding the limit as \(x\) approaches \(\infty\)

Since \(x>\=0\) when \(x\) approaches \(\infty\), we will use \(-1\) part. Hence, the limit as \(x\) approaches \(\infty\) for the function \(f(x) = -1\) is -1.

Finding the limit as \(x\) approaches 0 from the left, denoted as \(x \rightarrow 0^{-}\)

As \(x\) approaches 0 from the left (\(x<0\)), we use the function \(f(x) = 1 /x\). The limit is \(\infty\).

Finding the limit as \(x\) approaches 0 from the right, denoted as \(x \rightarrow 0^{+}\)

As \(x\) approaches 0 from the right (\(x\>=0\)), we use the function \(f(x) = -1\). The limit is -1.

Key concepts

Finding Limits

Understanding the concept of finding limits is fundamental in calculus. Limits describe the behavior of a function as its input approaches some value. In mathematical terms, we say that the limit of a function \( f(x) \) as \( x \) approaches a value \( a \) is \( L \) if the values of \( f(x) \) get arbitrarily close to \( L \) as \( x \) gets arbitrarily close to \( a \). This concept does not necessarily concern itself with the function's value at \( a \), but rather where the function is heading as it approaches from either side.
To calculate a limit, we can use various methods, such as direct substitution, factoring, rationalization, or even graphical analysis. In the given exercise, the function \( f(x) \) changes its formula depending on whether \( x \) is less than or greater than 0. Therefore, we must evaluate the limit using the appropriate piece of the function for the value that \( x \) is approaching.

Limits at Infinity

When evaluating limits at infinity, we're interested in the behavior of a function as its input grows without bound in the positive or negative direction. For functions that behave differently on different intervals, we have to consider the piece of the function that corresponds to the direction from which \( x \) is approaching infinity.
In the provided exercise, when \( x \) approaches negative infinity (\( x \to -\infty \)), we use the portion of the function that applies to negative values of \( x \), which is \( \frac{1}{x} \). This function approaches 0 as \( x \) becomes increasingly negative. Similarly, when \( x \) approaches positive infinity (\( x \to \infty \)), the relevant part of the function is the constant \( -1 \), because \( x \) is greater than or equal to 0 in this range, and as such, the limit is \( -1 \).

Left-Hand Limit

A left-hand limit specifically refers to the limit of a function as the input approaches a certain value from the left side, or from values less than the value in question. In mathematical notation, this is denoted as \( x \to c^- \). It is a crucial concept for understanding the behavior of a function at a specific point, especially for piecewise functions or functions that are not continuous.
For example, in our step-by-step exercise, to find the left-hand limit as \( x \) approaches 0 (\( x \to 0^- \)), we look at the function as \( x \) approaches 0 from values less than 0. In this case, the left-hand limit of \( f(x) = \frac{1}{x} \) as \( x \) approaches 0 is \( \infty \), indicating that the function increases without bound as it approaches 0 from the left.

Right-Hand Limit

Conversely, a right-hand limit describes the behavior of a function as its input approaches a value from the right, or from values greater than the value in question. This is notated as \( x \to c^+ \). It's equally important as the left-hand limit in determining the behavior of a function at a discontinuity or a point of interest.
In the textbook example provided, the right-hand limit as \( x \) approaches 0 (\( x \to 0^+ \)) is found by looking at the function for values of \( x \) greater than 0. Here, the function \( f(x) \) is simply \( -1 \), and remains unchanged as \( x \) approaches 0 from the right. Therefore, the right-hand limit is \( -1 \).