Problem 14 - APEX Calculus

Question

Approximate the given limits both numerically and graphically. $$ \begin{array}{l} \lim _{x \rightarrow 3} f(x), \text { where } \\ f(x)=\left\\{\begin{array}{cl} x^{2}-x+1 & x \leq 3 \\ 2 x+1 & x>3 \end{array}\right. \end{array} $$

Step-by-Step Solution

Answer

The limit \( \lim_{x \to 3} f(x) \) is 7.

Step 1: Analyze the Piecewise Function

The function \( f(x) \) is given as a piecewise function: \( f(x) = x^2-x+1 \) if \( x \leq 3 \) and \( f(x) = 2x+1 \) if \( x>3 \). We need to check the behavior of the function from both sides of \( x=3 \).

Step 2: Evaluate the Left-Hand Limit

To find \( \lim_{x \to 3^-} f(x) \), use the piece of the function where \( x \leq 3 \). Evaluate \( f(x) = x^2 - x + 1 \) at \( x=3 \):\[ f(3) = 3^2 - 3 + 1 = 9 - 3 + 1 = 7. \]

Step 3: Evaluate the Right-Hand Limit

To find \( \lim_{x \to 3^+} f(x) \), use the piece of the function where \( x > 3 \). Evaluate \( f(x) = 2x + 1 \) as \( x \) approaches 3:\[ f(3) = 2(3) + 1 = 6 + 1 = 7. \]

Step 4: Verify Both Limits Consistently

Both the left-hand and right-hand limits are equal, \( \lim_{x \to 3^-} f(x) = 7 \) and \( \lim_{x \to 3^+} f(x) = 7 \). Thus, the limit at \( x = 3 \) is:\[ \lim_{x \to 3} f(x) = 7. \]

Step 5: Graphical Approximation

Plot the piecewise function using a graph, where \( x^2-x+1 \) is used for \( x \leq 3 \) and \( 2x+1 \) for \( x > 3 \). Observe the continuity at \( x = 3 \) where both function segments meet at the point (3, 7).

Key Concepts

Piecewise Functions

A piecewise function is a type of function defined by multiple sub-functions, each of which applies to a certain interval of the domain. In simpler terms, it is a function that behaves differently based on the input value.
For the function \( f(x) \) given in this exercise, we have:

• \( f(x) = x^2 - x + 1 \) for \( x \leq 3 \)
• \( f(x) = 2x + 1 \) for \( x > 3 \)

This means that for every input \( x \) less than or equal to 3, the function uses the formula \( x^2 - x + 1 \). If \( x \) is greater than 3, the formula becomes \( 2x + 1 \).
Understanding how piecewise functions work is crucial in evaluating limits as each piece needs to be considered according to its valid domain.

Left-Hand Limit

The left-hand limit analyzes the value a function approaches as the input nears a particular point from the left side. In our exercise, we find the left-hand limit of \( f(x) \) at \( x = 3 \):

• For \( x \leq 3 \), use the function \( f(x) = x^2 - x + 1 \).
• Substitute \( x = 3 \) into the function: \( f(3) = 3^2 - 3 + 1 = 7 \).

This calculation confirms that as \( x \) approaches 3 from the left, \( f(x) \) approaches 7.
When evaluating limits, the behavior of the function from the left must smoothly transition to the function value at the point in focus.

Right-Hand Limit

The right-hand limit examines how a function approaches a specific input value from values greater than the target input. For our exercise at \( x = 3 \):

• Since \( x > 3 \), we use \( f(x) = 2x + 1 \).
• To approximate the limit as \( x \rightarrow 3^{+} \), evaluate \( f(x) \) at \( x = 3 \): \( f(3) = 2(3) + 1 = 7 \).

As \( x \) approaches 3 from the right, \( f(x) \) similarly converges to 7.
The importance of the right-hand limit reveals whether the function maintains consistency approaching a value from the right side relative to the left side and contributes to determining overall continuity.

Graphical Approximation

Graphing a piecewise function helps in visualizing its behavior around a point of interest. In this particular function, two distinct linear segments are crucial.

• Plot \( y = x^2 - x + 1 \) for \( x \leq 3 \), showing a curve segment.
• Plot \( y = 2x + 1 \) for \( x > 3 \), depicting a straight line.

When graphed, observe that at \( x = 3 \), the end of the curve and the start of the line both meet at the point (3, 7).
This visualization further confirms the calculated limits and illustrates that the function is continuous at \( x = 3 \) as both left-hand and right-hand sides converge smoothly.