Problem 72 - Calculus: Early Transcendentals

Question

One-sided limits Let $$g(x)=\left\\{\begin{array}{ll}5 x-15 & \text { if } x<4 \\\\\sqrt{6 x+1} & \text { if } x \geq 4\end{array}\right.$$ Compute the following limits or state that they do not exist. 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

Answer: The overall limit of the piecewise function \(g(x)\) as \(x\) approaches \(4\) is \(5\).

Step 1: Left-hand limit (\(x \rightarrow 4^{-}\))

Since we are asked to find the limit as \(x\) approaches \(4\) from the left (\(x \rightarrow 4^{-}\)), we will use the expression for \(g(x)\) when \(x<4\), which is \(g(x)=5x-15\). To compute the limit, we simply plug \(x=4\) into this expression: $$\lim_{x\rightarrow 4^{-}} g(x) = 5(4) - 15.$$

Step 2: Compute the left-hand limit

Solving for the left-hand limit, we get: $$\lim_{x\rightarrow 4^{-}} g(x) = 20 - 15=5.$$

Step 3: Right-hand limit (\(x \rightarrow 4^{+}\))

Now we need to find the limit as \(x\) approaches \(4\) from the right (\(x \rightarrow 4^{+}\)). This means we will use the expression for \(g(x)\) that is valid when \(x \geq 4\), which is \(g(x)=\sqrt{6x+1}\). To compute the limit, substitute \(x=4\) into this expression: $$\lim_{x\rightarrow 4^{+}} g(x) = \sqrt{6(4)+1}.$$

Step 4: Compute the right-hand limit

Solving for the right-hand limit, we get: $$\lim_{x\rightarrow 4^{+}} g(x) = \sqrt{24 + 1} = \sqrt{25}=5.$$

Step 5: Compute the overall limit

Since we have found that both left-hand and right-hand limits are equal (\(\lim_{x\rightarrow 4^{-}} g(x) = \lim_{x\rightarrow 4^{+}} g(x) = 5\)), we can conclude that the overall limit exists and is equal to \(5\): $$\lim_{x \rightarrow 4} g(x) = 5.$$ To summarize our findings: a. \(\lim _{x \rightarrow 4} g(x) = 5\) b. \(\lim _{x \rightarrow 4^{+}} g(x) = 5\) c. \(\lim _{x \rightarrow 4} g(x) = 5\)

Key Concepts

Limit from the left

A one-sided limit from the left refers to the behavior of a function as the input values approach a specific point from values less than that point. Essentially, you look at the values of the function that are "left" of the number in question. In mathematical notation, this is expressed as \( \lim_{x \to a^-} f(x) \). Here, it indicates that we are finding the limit as \( x \) approaches \( a \) from the left side.

In the given example, for the piecewise function \( g(x) \), the expression used when \( x < 4 \) is \( g(x) = 5x - 15 \). To find \( \lim_{x \to 4^-} g(x) \), we substitute \( x = 4 \) into this part of the function, leading to a simple calculation:

• Calculate \( 5 \times 4 - 15 \).
• The result is \( 5 \).

This indicates that as \( x \) gets closer and closer to \( 4 \) from the left, \( g(x) \) approaches \( 5 \). This step tells us how the function behaves just as \( x \) reaches \( 4 \) coming from values less than \( 4 \).

Limit from the right

The limit from the right concerns the behavior of a function as it approaches a certain value from the greater side—that is, from numbers larger than the point of interest. In notations, this is shown as \( \lim_{x \to a^+} f(x) \). So when observing from the right, it involves looking at how the function values tend when approached from that direction.

For our function \( g(x) \), the expression applicable when \( x \geq 4 \) is \( g(x) = \sqrt{6x + 1} \). To determine \( \lim_{x \to 4^+} g(x) \):

• Substitute 4 into \( \sqrt{6(4) + 1} \).
• This simplifies to \( \sqrt{24 + 1} = \sqrt{25} = 5 \).

Thus, as \( x \) approaches \( 4 \) from the right, the value of the function \( g(x) \) approaches \( 5 \). This helps us understand how the function behaves exactly when \( x \) is just beyond \( 4 \).

Piecewise function

A piecewise function is defined by different expressions for different intervals of the input values, i.e., it has various output equations depending on which part of the domain you are considering.

In our case, the function \( g(x) \) is split into two parts:

• For \( x < 4 \), the function is \( g(x) = 5x - 15 \).
• For \( x \geq 4 \), the function is \( g(x) = \sqrt{6x + 1} \).

This nature of a piecewise function is crucial for evaluating limits, particularly at the points where the definition changes—in this instance, at \( x = 4 \). At such junctions, determining the one-sided limits helps in finding whether the overall limit exists.

For the given problem, since both left-hand and right-hand limits at \( x = 4 \) equal \( 5 \), the overall limit \( \lim_{x \to 4} g(x) \) is \( 5 \). Evaluating piecewise functions requires analyzing each segment separately where the function might change and observe if these segments lead to a singular behavior at the point of interest.