Problem 40 - 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. \(\lim _{x \rightarrow 4} f(x)\) $$ f(x)=\left\\{\begin{array}{ll} 2-x & \text { if } x<4 \\ 2 x-10 & \text { if } x \geq 4 \end{array}\right. $$

Step-by-Step Solution

Answer

a: -2, b: -2, c: -2.

Step 1: Identify the Left-Hand Limit

To find \( \lim_{x \to 4^-} f(x) \), we examine the function piece that applies when \( x < 4 \). This is \( f(x) = 2 - x \). So we compute the limit: \( \lim_{x \to 4^-} f(x) = \lim_{x \to 4^-} (2 - x) \).

Step 2: Evaluate Left-Hand Limit

Substitute 4 into the expression \( 2-x \). \( \lim_{x \to 4^-} (2 - x) = 2 - 4 = -2 \).

Step 2: Identify the Right-Hand Limit

To find \( \lim_{x \to 4^+} f(x) \), we examine the function piece that applies when \( x \geq 4 \). This is \( f(x) = 2x - 10 \). So we compute the limit: \( \lim_{x \to 4^+} f(x) = \lim_{x \to 4^+} (2x - 10) \).

Step 4: Evaluate Right-Hand Limit

Substitute 4 into the expression \( 2x - 10 \). \( \lim_{x \to 4^+} (2 \times 4 - 10) = 8 - 10 = -2 \).

Step 3: Determine the Two-Sided Limit

The two-sided limit \( \lim_{x \to 4} f(x) \) exists if and only if the left-hand limit and the right-hand limit are equal. We have: \( \lim_{x \to 4^-} f(x) = -2 \) and \( \lim_{x \to 4^+} f(x) = -2 \). Since both are equal, \( \lim_{x \to 4} f(x) = -2 \).

Key Concepts

Understanding Limits

The concept of limits is crucial in calculus and helps us understand the behavior of functions as they approach a specific point. Essentially, the limit tells us what value a function is approaching as the input gets closer and closer to a particular number. Limits help in checking the continuity of functions at particular points and are foundational for calculus concepts such as derivatives and integrals.

When we look at a function's limit as \( x \) approaches a number \( a \), we are interested in seeing what value the function \( f(x) \) is closer to as \( x \) nears \( a \). If \( f(x) \) becomes nearly equal to a certain number as \( x \) gets close to \( a \), then we say that the limit of \( f(x) \) as \( x \) approaches \( a \) is that number.

• Limits help determine the behavior of functions near specified points.
• Understanding both sides of a point helps form a full view of a limit.
• Without limits, calculus wouldn’t be able to move from average rates to instantaneous rates of change.

Left-Hand Limit

In the context of limits, the left-hand limit involves observing a function as the variable approaches the target value from the left-hand side. This means we approach the point of interest through values that are less than the point itself.

For a function \( f(x) \), \( \lim_{x \to a^-} f(x) \) represents the left-hand limit of \( f(x) \) as \( x \) approaches \( a \). Effectively, it tells us what value _f(x)_ is getting closer to as _x_ approaches _a_ from values less than _a_. This approach is essential to understand how functions behave just before reaching a threshold.

• The left-hand limit considers only the values of \( x \) smaller than \( a \).
• It helps define how a function behaves right before a key point.
• Understanding both the left and right limits aids in determining function continuity.

Right-Hand Limit

The right-hand limit concerns observing a function as the variable approaches a specific value from the right-hand side. This direction means we're focusing on the values greater than the target value but getting closer to it.

For a function \( f(x) \), \( \lim_{x \to a^+} f(x) \) signifies the right-hand limit of \( f(x) \) as \( x \) nears \( a \) from the right. This is about understanding the function's behavior immediately after reaching the threshold. Just like the left-hand limit, this specific approach aids in painting a fuller picture of a function’s character.

• The right-hand limit considers only values of \( x \) greater than \( a \).
• It highlights the function's behavior immediately after a key point.
• When combined with the left-hand limit, it helps judge if a function is continuous at a point.

Two-Sided Limit

A two-sided limit is a comprehensive look at how both sides of the limit (left-hand and right-hand) converge on the same value. For a limit to exist at a given point, both the left-hand and right-hand limits must agree, meaning they approach the same number.

For a function \( f(x) \), \( \lim_{x \to a} f(x) \) represents the two-sided limit as \( x \) approaches \( a \). This two-sided nature means we’re examining \( f(x) \) from both directions, crafting a joint verdict of the function’s tendencies as \( x \) closes in on \( a \).

• A two-sided limit only exists if both left and right-hand limits are equal.
• This includes both directions: approaching \( a \) from lesser and greater than values.
• If different, the function is said to be discontinuous at that point.