Problem 62 - Precalculus with Limits

Question

In Exercises 55-62, graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits. \\[ \lim_{x \to 0} f(x)\ \textrm{where}\ f(x) = \begin{cases} 4-x^2, & \quad x\leq0\\\ x+4, & \quad x>0 \end{cases} \\]

Step-by-Step Solution

Answer

The left-hand limit as \( x \) approaches 0 is 4, and the right-hand limit as \( x \) approaches 0 is also 4. Therefore, \( \lim_{x \to 0} f(x) \) equals 4.

Step 1: Graphical Representation of the function

For \(x \leq 0, f(x) = 4 - x^2\). Plot the curve for this function in the respective interval. And for \( x > 0, f(x) = x + 4 \). Similarly, plot the curve for this function in the respective interval. Observe where these two segments of the graph meet up or don't meet.

Step 2: Evaluate the Left-Hand Limit

Evaluate the left-hand limit, \( \lim_{x \to 0^-} f(x) \) by substituting \( x \) with a value slightly less than 0 into the function, in this case \( f(x) = 4 - x^2 \).

Step 3: Evaluate the Right-Hand Limit

Evaluate the right-hand limit, \( \lim_{x \to 0^+} f(x) \) by substituting \( x \) with a value slightly more than 0 into the function, in this case \( f(x) = x + 4 \).

Step 4: Compare the One-Sided Limits

If both the left-hand and right-hand limits equal to the same value, then the overall limit of the function as \( x \) approaches 0 exists and is equal to this value. If not, the limit does not exist.

Key Concepts

One-sided limits

When evaluating the limit of a function as it approaches a specific point, we often need to consider one-sided limits first. A one-sided limit analyzes the behavior of the function as it approaches the point from one direction only: the left or the right.

• The **left-hand limit** is denoted as \( \lim_{x \to c^-} f(x) \), where \( x \) approaches the point \( c \) from values less than \( c \).
• The **right-hand limit** is denoted as \( \lim_{x \to c^+} f(x) \), where \( x \) approaches \( c \) from values greater than \( c \).

In the exercise, you found the left-hand limit \( \lim_{x \to 0^-} f(x) \) by evaluating \( 4 - x^2 \) as \( x \) approaches 0 from the left.
You also evaluated the right-hand limit \( \lim_{x \to 0^+} f(x) \) using the function \( x + 4 \) as \( x \) moves towards 0 from the right.
These one-sided limits help us determine if the overall limit exists at that point by comparing the two results.

Piecewise functions

A piecewise function is a type of function defined by different expressions depending on the interval of the input. Such functions offer a flexible approach to defining diverse behaviors, which can be represented through various segments.

• In the exercise, the function \( f(x) \) is defined by two separate formulas:
• \( f(x) = 4 - x^2 \) for \( x \leq 0 \)
• \( f(x) = x + 4 \) for \( x > 0 \)

Depending on the value of \( x \), one part of the function applies, defining different outcomes in different regions. This segmented structure requires careful evaluation at points where the expression changes, ensuring continuity or identifying any discontinuities.
Evaluating limits for piecewise functions usually involves checking the behavior at the boundaries, like at \( x = 0 \) in this case.

Graphical analysis

Graphical analysis is a powerful tool in understanding function behavior and their limits. By sketching the function's graph, we get visual insights into the continuity and points of interest in piecewise functions.
In the exercise, graphing \( f(x) \) involves two components:

• For \( x \leq 0 \), the curve follows \( 4 - x^2 \). This is a downward-opening parabola.
• For \( x > 0 \), it follows \( x + 4 \), a straight line with a positive slope.

The graphical representation helps to observe where the two parts of the function meet. By plotting these sections, you can see if they connect seamlessly or if there's a gap or jump at the point.
For \( \lim_{x \to 0} f(x) \), examining the graph around \( x = 0 \) provides a visual explanation of whether the limit exists by seeing if the left and right parts align.