Problem 28 - Calculus of a Single Variable

Question

In Exercises 27 and 28, sketch a graph of a function that satisfies the given values. (There are many correct answers.) $$ \begin{array}{l}{f(-2)=0} \\ {f(2)=0} \\ {\lim _{x \rightarrow-2} f(x)=0} \\\ {\lim _{x \rightarrow 2} f(x) \text { does not exist. }}\end{array} $$

Step-by-Step Solution

Answer

One possible graph of the function could be a function that ascends smoothly from negative infinity to touch the point (-2,0), remains constant up to (2,0) and then jumps to a non-zero value at x=2.

Step 1: Identify given properties of function f

We are given that the function satisfies the values \(f(-2)=0\), \(f(2)=0\), \(\lim _{x \rightarrow-2} f(x)=0\) and \(\lim _{x \rightarrow 2} f(x)\) does not exist.

Step 2: Sketch the graph

As \(f(-2) = 0\) and \(f(2) = 0\), we can start by marking these two points, (-2, 0) and (2, 0), on the graph. Since \(\lim _{x \rightarrow-2} f(x)=0\), it requires that the function f(x) approaches 0 as x approaches -2 from both right and left. This suggests that the graph would appear to 'touch' the x-axis at x=-2. However, since \(\lim _{x \rightarrow 2} f(x)\) does not exist, the graph must behave differently at x=2 from the left and from the right, i.e., the function doesn't settle on a particular value near x=2. A typical example would be a hole or a jump in the graph of f(x) at x=2.

Step 3: Finalize the Graph

There are many possible final sketches for this problem, but we choose one where the function f(x) approaches and 'touches' the x-axis at x=-2 and has a discontinuity (in this case a jump) at x=2.

Key Concepts

Discontinuity

In calculus, a function is said to have a discontinuity at a point where it is not smooth or uninterrupted. At a discontinuous point, there could be a sudden jump, a hole, or a cusp in the graph. Discontinuities are critical in understanding how functions behave because they indicate points where the function does not smoothly transition.
When we say that a graph has a discontinuity at a point, it means one of several things could be happening, such as:

• A jump discontinuity, where the function has a defined value on either side of the point but jumps to a different value.
• A removable discontinuity, often represented as a hole in the graph where the function is not defined, yet it appears as though the limit exists.
• An infinite discontinuity, where the function heads towards infinity as it approaches the point in question.

In the exercise, the function exhibits a discontinuity at x=2. This is shown by the fact that the limit does not exist as x approaches 2, indicating a jump or hole in the graph.

Function Graph

The graph of a function provides a visual representation of the function's values across different inputs. It's like a map that shows the terrain of a function, where points along the curve tell us what the function's output is for any given input.
When sketching the graph of a function, it is vital to pinpoint specific known values and behaviors, such as:

• The function's defined values at specific points—such as nodes or zeros where the graph crosses the x-axis, like f(-2) = 0 and f(2) = 0 in the exercise.
• Important points of behavior, like where the function approaches a value, shown by limits.
• Asymptotic behavior where the function draws closer to a line but never quite reaches it, often seen in rational functions.

For the given exercise, starting with points (-2, 0) and (2, 0) helps in sketching the graph. Seeing these points on a graph lets us place anchors that guide our drawing and help predict the function's trajectory in between these points and beyond.

Limit Does Not Exist

In calculus, the concept of limits is essential for understanding how functions behave as they approach specific points. We say that the limit of a function does not exist at a certain point if the function doesn't approach a single, finite value as its input gets close to that point.
There are several reasons why a limit may not exist for a function at a point:

• The function might approach different values from the left side compared to the right side, resulting in what's known as a jump discontinuity.
• The function could oscillate infinitely without settling near a single value. An example of this is trigonometric functions like sine or cosine at some critical angles.
• The function might head towards infinity, like how 1/x behaves as x approaches 0.

In the exercise, it is noted that \(\lim_{x \to 2} f(x)\) does not exist, indicating such irregular behavior at x=2—a clear sign of discontinuity that could manifest as a jump in the function.