Problem 39 - Brief Applied Calculus

Question

For each piecewise linear function, find: a. \(\lim _{x \rightarrow 4^{-}} f(x)\) b. \(\lim _{x \rightarrow 4^{+}} f(x)\) c. \(\lim _{x \rightarrow 4} f(x)\) $$ f(x)=\left\\{\begin{array}{ll} 2-x & \text { if } x \leq 4 \\ x-6 & \text { if } x>4 \end{array}\right. $$

Step-by-Step Solution

Answer

The limits are: a. \(-2\), b. \(-2\), c. \(-2\).

Step 1: Evaluate Left-Hand Limit

For \( \lim _{x \rightarrow 4^{-}} f(x) \), consider the function where \( x \leq 4 \): \[ f(x) = 2 - x \] Substitute \( x = 4 \) into this expression: \[ f(4) = 2 - 4 = -2 \] Thus, \( \lim _{x \rightarrow 4^{-}} f(x) = -2 \).

Step 2: Evaluate Right-Hand Limit

Next, find \( \lim _{x \rightarrow 4^{+}} f(x) \) using the function for \( x > 4 \): \[ f(x) = x - 6 \] Substitute \( x = 4 \) into this expression, noting that it represents values just larger than \( 4 \): \[ f(4) = 4 - 6 = -2 \] Thus, \( \lim _{x \rightarrow 4^{+}} f(x) = -2 \).

Step 3: Evaluate Two-Sided Limit

To determine if \( \lim _{x \rightarrow 4} f(x) \) exists, compare the left-hand and right-hand limits. Since both \( \lim _{x \rightarrow 4^{-}} f(x) \) and \( \lim _{x \rightarrow 4^{+}} f(x) \) equal \(-2\), the two-sided limit is:\[ \lim _{x \rightarrow 4} f(x) = -2 \].

Key Concepts

Limits in Calculus

Limits are fundamental concepts in calculus and play a crucial role when dealing with functions, especially piecewise functions. They help us understand how a function behaves as it approaches a particular input value. This is especially useful when a function itself is not specified at that point, or if it's not continuous. A limit examines the approaching value, showing the trend that the function values are getting closer to as they near a certain point.

In dealing with limits, we consider three common types: the left-hand limit, the right-hand limit, and the two-sided limit. Each type provides vital insights into the behavior of piecewise functions as they approach particular points from different directions. With these concepts, one can determine if there is a consistent trend (i.e., the same directional behavior) in the function as it nears a specific value. Limits provide a clearer picture of the function's continuity and help identify any points of discontinuity.

Left-Hand Limit

The left-hand limit examines how a function behaves as the input value approaches a specific point from the left. This means we're interested in what the function values do as they get infinitely close to this point from values smaller than it.

For example, let's consider the left-hand limit problem:

• The function defined was different for values of \(x\) less than or equal to 4: \(f(x) = 2 - x\).
• To find the left-hand limit \(\lim _{x \rightarrow 4^{-}} f(x)\), you plug in the value \(x=4\) into the function \(2-x\), which results in \(-2\).

This calculation shows that as \(x\) approaches 4 from the left, the function gets closer to -2.
This method is effective for piecewise functions, where the function definition can change suddenly. The left-hand limit shows the behavior just before that change.

Right-Hand Limit

The right-hand limit involves examining a function as the input approaches a specific value from the right. This focuses on values coming from larger than the specified point, highlighting how the function behaves as it nears that point.

In our exercise:

• The definition for values greater than 4 is \(f(x) = x - 6\).
• To find \(\lim _{x \rightarrow 4^{+}} f(x)\), use the expression \(x-6\) with \(x=4\), resulting in \(-2\).

This indicates that as the function approaches \(x = 4\) from the right, it tends towards -2.

Right-hand limits are particularly useful in situations where functions exhibit a different trend just past a point compared to just before it. This is quintessential in piecewise functions, where behaviors can completely change at boundary values.

Two-Sided Limit

A two-sided limit is determined by considering both the left-hand and right-hand limits of a function as the input approaches a certain point. If both one-sided limits agree, then the two-sided limit at that point exists and is equal to these matching one-sided limits.

In our exercise, we want to determine if \(\lim _{x \rightarrow 4} f(x)\) exists:

• The left-hand limit is \(-2\), as previously calculated using \(f(x) = 2 - x\).
• The right-hand limit is also \(-2\), obtained from \(f(x) = x - 6\).

Since both one-sided limits equal \(-2\), the two-sided limit is \(-2\).

This means that as \(x\) gets closer to 4 from either direction, the function value is approaching \(-2\). Ensuring that one-sided limits are equal is crucial for the existence of a two-sided limit and often reflects continuity in function behavior.