Question

Use a graphing utility to graph the function. From the graph, estimate $$ \lim _{x \rightarrow 0^{+}} f(x) \text { and } \lim _{x \rightarrow 0^{-}} f(x) $$ Is the function continuous on the entire real number line? Explain. $$ f(x)=\frac{\left|x^{2}-4\right| x}{x+2} $$

Short answer

Use a graphing utility to plot the function. Estimate the right-side limit by observing the y-value as x approaches 0 from the right on the graph, and estimate the left-side limit by observing the y-value as x approaches 0 from the left. Finally, by scrutinizing the entire graph, we can determine whether the function is continuous across the entire real number line.

Step-by-step solution

Graphing the function

We use a graphing utility to plot \( f(x)=\frac{\left|x^{2}-4\right| x}{x+2} \).

Estimate the right-side and left-side limit as x approach 0

Observing the graph, we can notice that as x approaches 0 from the right (\( x \rightarrow 0^{+} \)), the function's output appears to approach a certain y-value. The same process applies as x approaches 0 from the left side (\( x \rightarrow 0^{-} \)). These are the right-side limit and left-side limit, respectively.

Determine the continuity on the entire real number line

The function is continuous over the entire real number line when there are no sudden jumps, breaks, or holes in the function when graphed. So by analyzing the graph, we can determine if \( f(x) \) meets these conditions.

Key concepts

Graphing Functions

Understanding the concept of graphing functions is crucial for a variety of calculus problems, including finding limits. Graphing a function provides a visual representation that helps to describe its behavior over a range of values. It involves plotting points on a coordinate plane where the horizontal axis (x-axis) represents the input values, and the vertical axis (y-axis) represents the output or function values (f(x)).

When graphing the function f(x) given in the exercise, f(x) = \(\frac{\left|x^2-4\right|x}{x+2}\), one would calculate the y-values for selected x-values and plot these points on the graph. Because the function includes an absolute value, it will reflect across the x-axis for negative values inside the absolute value, creating a distinct 'V' shape at the point x=2. The approach of calculating and plotting points is especially helpful to observe the function's behavior near crucial points, such as x=0 and x=-2 in this case.

Right-Side Limit

The right-side limit of a function as x approaches a certain value involves looking at the behavior of the function from values greater than the point of interest, moving closer to the point itself. It is denoted as \( \lim _{x \rightarrow c^{+}} f(x) \), where 'c' is the value x is approaching, and the '+' sign indicates that x is approaching from the right. In the exercise, the right-side limit as x approaches zero is what we observe when the input values become infinitesimally small, but positive.

For the function \( f(x) = \frac{\left|x^2-4\right|x}{x+2} \), graphing it shows how the function's value behaves as x approaches 0 from positive values. Understanding this helps to anticipate the function's behavior at points that we can't necessarily evaluate directly, like when we approach a discontinuity from a specific direction.

Left-Side Limit

Conversely, the left-side limit refers to the behavior of a function as the input values approach a certain point from values less than that point. The mathematical notation for this concept is \( \lim _{x \rightarrow c^{-}} f(x) \), with the '-' sign indicating that x is approaching from the left side. For the function provided in the exercise, assessing the left-side limit at x=0 involves looking at how the function behaves as x gets increasingly negative, closer to zero.

When the function f(x) is graphed, observing the trend as x approaches 0 from the left side can reveal whether the function's values are converging to a specific y-value or if they exhibit a different trend. This aspect of limits is essential in determining whether a function is continuous at certain points or within certain intervals.

Continuity of Functions

The concept of continuity relates to whether a function has any interruptions, jumps, or breaks over its domain. A function is continuous at a point if the right-side limit, the left-side limit, and the actual value of the function at that point all exist and are equal. The notation f(c) = \( \lim _{x \rightarrow c} f(x) \) summarizes continuity at a specific point 'c'.

The graphical approach helps determine continuity as we can visually assess whether the plotted points connect to form an unbroken curve. The function given, \( f(x) = \frac{\left|x^2-4\right|x}{x+2} \), when graphed, should be evaluated for continuity by checking for gaps, jumps, or vertical asymptotes. In this exercise, we are particularly interested in its behavior around x=0 and across the entire real number line. If the function passes the 'pencil test', where a pencil can draw the function without lifting from the paper, then it suggests the function is continuous across its domain.