Problem 14 - Calculus and its applications

Question

Consider the function g given by $$ g(x)=\left\\{\begin{array}{ll} x+6, & \text { for } x<-2, \\ -\frac{1}{2} x+1, & \text { for } x \geq-2. \end{array}\right. $$ If a limit does not exist, state that fact. Find (a) \(\lim _{x \rightarrow 4^{-}} g(x)\) (b) \(\lim _{x \rightarrow 4^{+}} g(x)\) (c) \(\lim _{x \rightarrow 4} g(x)\).

Step-by-Step Solution

Answer

All limits are -1; thus, \( \lim_{x \to 4} g(x) = -1.\)

Step 1: Determine the Relevant Piece for \( \lim _{x \rightarrow 4^{-}} g(x) \)

For \( x \to 4^- \), we are considering values of \( x \) that approach 4 from the left. Since 4 is greater than -2, the relevant piece of the piecewise function is the second case: \( g(x) = -\frac{1}{2}x + 1. \) Thus, we need to find \( \lim_{x \to 4^-} \left(-\frac{1}{2}x + 1\right).\)

Step 2: Compute \( \lim _{x \rightarrow 4^{-}} g(x) \)

Plug \( x = 4 \) into \( g(x) = -\frac{1}{2}x + 1 \):\[ g(4) = -\frac{1}{2}(4) + 1 = -2 + 1 = -1. \]Therefore, \( \lim_{x \rightarrow 4^{-}} g(x) = -1. \)

Step 3: Determine the Relevant Piece for \( \lim _{x \rightarrow 4^{+}} g(x) \)

For \( x \to 4^+ \), we are considering values of \( x \) that approach 4 from the right. Since 4 is greater than -2, the relevant piece of the piecewise function is again the second case: \( g(x) = -\frac{1}{2}x + 1. \) Thus, we need to find \( \lim_{x \to 4^+} \left(-\frac{1}{2}x + 1\right).\)

Step 4: Compute \( \lim _{x \rightarrow 4^{+}} g(x) \)

Plug \( x = 4 \) into \( g(x) = -\frac{1}{2}x + 1 \):\[ g(4) = -\frac{1}{2}(4) + 1 = -2 + 1 = -1. \]Therefore, \( \lim_{x \rightarrow 4^{+}} g(x) = -1. \)

Step 5: Determine \( \lim_{x \rightarrow 4} g(x) \)

To find \( \lim_{x \rightarrow 4} g(x) \), we must check if the left-hand limit and the right-hand limit at \( x = 4 \) are equal. Both \( \lim_{x \rightarrow 4^{-}} g(x) \) and \( \lim_{x \rightarrow 4^{+}} g(x) \) are equal to -1. Since these two one-sided limits are equal, \( \lim_{x \rightarrow 4} g(x) = -1. \)

Key Concepts

Piecewise Functions

In calculus, piecewise functions are fascinating because they're like a map with different routes based on your starting point! A piecewise function is composed of multiple sub-functions. Each of these sub-functions applies to different intervals of the main function's domain.
For example, the function \( g(x) \) in this exercise is defined by two pieces: \( x + 6 \) for \( x < -2 \) and \(-\frac{1}{2}x + 1 \) for \( x \geq -2 \). Depending on whether \( x \) is less than or greater than/equal to -2, different formulas determine the output of \( g(x) \).
Key characteristics of piecewise functions include:

• **Domain Segmentation**: Different rules apply to different segments.
• **Discontinuity Potential**: Points where the pieces meet can lead to different behaviors like jumps or holes.
• **Flexibility**: Allows modeling of complex, real-world situations more accurately.

Understanding piecewise functions is crucial for analyzing scenarios where behavior changes at certain thresholds.

One-sided Limits

One-sided limits are a handy tool for studying how a function behaves as it approaches a certain point from one direction—either from the left or the right.

When you consider the left-hand limit, denoted as \( \lim_{x \rightarrow c^{-}} f(x) \), you're looking at the values \( x \) that approach \( c \) from the left. The right-hand limit, written as \( \lim_{x \rightarrow c^{+}} f(x) \), examines \( x \) values approaching from the right.
In the exercise, evaluating \( \lim_{x \rightarrow 4^{-}} g(x) \) and \( \lim_{x \rightarrow 4^{+}} g(x) \) required us to determine which part of the piecewise function applied. Because \( g(x) \) switched formulas at \( x = -2 \), approaching 4 from either side used the same expression: \(-\frac{1}{2} x + 1 \).

Key points about one-sided limits include:

• **Directionality**: Specifies from which direction \( x \) approaches the point of interest.
• **Continuity Check**: If left and right-hand limits point to the same value, the overall limit exists at that point.
• **Detection of Discontinuities**: Different one-sided limits suggest a jump discontinuity or non-existence of a limit.

This concept is crucial for understanding the behavior of functions at boundaries and is often used in defining derivatives.

Continuous Functions

A continuous function is one where you can draw the graph without lifting your pen—this neat little idea keeps things predictable! In mathematical terms, a function \( f(x) \) is continuous at \( x = c \) if three conditions are satisfied:
1. \( f(c) \) is defined.
2. \( \lim_{x \to c} f(x) \) exists.
3. \( \lim_{x \to c} f(x) = f(c) \).

When dealing with piecewise functions, checking for continuity at the boundary is vital. In the exercise example, we wanted to check if \( g(x) \) was continuous at \( x = 4 \). Because both the left-hand and right-hand limits as \( x \) approaches 4 are equal to -1, and the function outputs \(-1\) at \( x = 4 \), \( g(x) \) remains continuous at this point.

Key takeaways about continuous functions include:

• **Smoothness**: Continuous functions have no breaks, jumps, or holes at the point of interest.
• **Interconnected Limits**: Equality of one-sided limits confirms continuity at that point.
• **Critical for Integration**: Many calculus operations, such as integration, rely on continuity to ensure the expected results are accurate and meaningful.

Understanding continuity helps ensure accurate analysis of functions across different intervals, particularly in applications involving real-world phenomena such as physics or economics.