Problem 31 - Calculus

Question

For the functions in Exercises \(29-31,\) use algebra to evaluate the limits \(\lim _{x \rightarrow a+} f(x), \lim _{x \rightarrow a-} f(x),\) and \(\lim _{x \rightarrow a} f(x)\) if they exist. Sketch a graph to confirm your answers. $$a=3, \quad f(x)=\left\\{\begin{array}{ll} x^{2}-2, & 0 < x < 3 \\ 2, & x=3 \\ 2 x+1, & 3 < x \end{array}\right.$$

Step-by-Step Solution

Answer

\(\lim_{x \to 3^-} f(x) = 7\), \(\lim_{x \to 3^+} f(x) = 7\), \(\lim_{x \to 3} f(x) = 7\), but \(f(3) = 2\).

Step 1: Analyze the Function Definition

The function \(f(x)\) is defined piecewise:- For \(0 < x < 3\), \(f(x) = x^2 - 2\).- At \(x = 3\), \(f(x) = 2\).- For \(x > 3\), \(f(x) = 2x + 1\).We need to evaluate the behavior of \(f(x)\) around \(x = 3\).

Step 2: Evaluate Right-Hand Limit (\(x \to 3^+\))

For \(x \to 3^+\), we use the third branch of the function, \(f(x) = 2x + 1\).- Plug in \(x = 3\) into the formula: \(2 \times 3 + 1 = 7\).Hence, \(\lim_{{x \to 3^+}} f(x) = 7\).

Step 3: Evaluate Left-Hand Limit (\(x \to 3^-\))

For \(x \to 3^-\), we use the first branch of the function, \(f(x) = x^2 - 2\).- Plug in \(x = 3\) into the formula: \(3^2 - 2 = 9 - 2 = 7\).Hence, \(\lim_{{x \to 3^-}} f(x) = 7\).

Step 4: Evaluate Two-Sided Limit (\(x \to 3\))

To evaluate \(\lim_{{x \to 3}} f(x)\), verify if the left-hand and right-hand limits are equal.Since both \(\lim_{{x \to 3^-}} f(x) = 7\) and \(\lim_{{x \to 3^+}} f(x) = 7\), the two-sided limit exists:\[\lim_{{x \to 3}} f(x) = 7\].

Step 5: Compare Limit and Function Value at \(x = 3\)

Since \(f(3) = 2\), the limit \(\lim_{{x \to 3}} f(x) = 7\) does not equal the function value at \(x = 3\). This indicates that \(f(x)\) is not continuous at \(x = 3\).

Step 6: Sketch the Graph for Confirmation

Create a piecewise graph to represent \(f(x)\):- For \(0 < x < 3\), draw the parabola \(y = x^2 - 2\).- At \(x = 3\), plot a point at \((3, 2)\) to represent \(f(3) = 2\).- For \(x > 3\), draw the linear function \(y = 2x + 1\).This graph should confirm that the limits from the left and right match the calculated values, but the value at \(x = 3\) is disconnected from these limits.

Key Concepts

Limits

In calculus, limits are essential tools to understand the behavior of functions as they approach a particular point. With piecewise functions, limits help in evaluating different behaviors of function segments around a specific value, in this case, around \(x = 3\). There are three types of limits:

• Right-Hand Limit (\(\lim_{{x \to a^+}} f(x)\)): This examines the value that \(f(x)\) approaches as \(x\) comes from values greater than \(a\). For the given exercise, the function for \(x > 3\) is \(2x + 1\). As \(x\) rides down to \(3\), the limit is calculated by substituting \(x = 3\) in this expression, giving us \(2 \times 3 + 1 = 7\).
• Left-Hand Limit (\(\lim_{{x \to a^-}} f(x)\)): This checks the value that \(f(x)\) nears as \(x\) approaches \(a\) from below. When \(x < 3\), the function follows \(x^2 - 2\). Plugging \(x = 3\) results in \((3^2 - 2) = 7\).
• Two-Sided Limit (\(\lim_{{x \to a}} f(x)\)): This limit exists if and only if both the right-hand and left-hand limits exist and equal each other. Thus, since both the left-hand and right-hand limits around 3 are \(7\), the two-sided limit becomes \(7\) as well.

Continuity

Continuity of a function at a particular point means that the function is smooth and unbroken at that point. A function \(f(x)\) is continuous at \(x=a\) if the following criteria are met:

• The function value \(f(a)\) exists.
• The limit \(\lim_{{x \to a}} f(x)\) exists.
• The limit equals the function value, i.e., \(\lim_{{x \to a}} f(x) = f(a)\).

Let's apply these conditions to our piecewise function at \(x = 3\). The limit we found was \(7\) as \(x\) approaches \(3\), while the function value \(f(3)\) is explicitly 2 (as given). Since \(7 eq 2\), \(f(x)\) fails the third condition. Therefore, \(f(x)\) is not continuous at \(x = 3\). This illustrates that sometimes functions can have a break or jump, highlighting real-world phenomena like sudden changes in data.

Graphical Analysis

Graphing the piecewise function provides a visual confirmation of the function's behavior and properties around the point in question.When sketching the graph, we plot according to each segment's definition:

• For \(0 < x < 3\), the parabola \(y = x^2 - 2\) represents the curve that approaches \(x = 3\) from the left. This segment will not include the endpoint at \(x = 3\).
• At \(x = 3\), plot a discrete point at \((3, 2)\) which represents the direct value given by the piecewise function at this point.
• For \(x > 3\), the line \(y = 2x + 1\) displays a linear increase as \(x\) moves beyond 3.

The resulting graph has a noticeable gap between the piecewise segments at \(x = 3\) where the function value is \(2\), but the left and right limits suggest it should be \(7\). This graphical gap confirms the discontinuity. Such analysis helps in visualizing theoretical concepts and confirms calculations made using algebraic methods.