Problem 1 - Brief Applied Calculus

Question

1-4. Complete the tables and use them to find the given limits. Round calculations to three decimal places. A graphing calculator with a TABLE feature will be very helpful. a. \(\lim _{x \rightarrow 2^{-}}(5 x-7)\) b. \(\lim _{x \rightarrow 2^{+}}(5 x-7)\) c. \(\lim _{x \rightarrow 2}(5 x-7)\)

Step-by-Step Solution

Answer

The limit is 3 for parts a, b, and c.

Step 1: Understanding Limit from the Left

To find the limit as \(x\) approaches 2 from the left of the function \(f(x) = 5x - 7\), observe the behavior of the function values as \(x\) gets closer to 2 by approaching from values less than 2. This is written as \(\lim_{x \rightarrow 2^-}(5x - 7)\).

Step 2: Create the Table for Left-hand Limit

Create a table with \(x\) values slightly less than 2, such as 1.9, 1.99, and 1.999. Calculate their corresponding \(f(x)\) values:- For \(x = 1.9\), \(f(1.9) = 5(1.9) - 7 = 2.5\)- For \(x = 1.99\), \(f(1.99) = 5(1.99) - 7 = 2.95\)- For \(x = 1.999\), \(f(1.999) = 5(1.999) - 7 = 2.995\)

Step 3: Observe the Limit from the Left-hand Side

As \(x\) approaches 2 from the left, the function values 2.5, 2.95, and 2.995 suggest that \(\lim_{x \rightarrow 2^-}(5x - 7)\) is approaching 3.

Step 4: Understanding Limit from the Right

To find the limit as \(x\) approaches 2 from the right of the function \(f(x) = 5x - 7\), observe the behavior for \(x\) values greater than 2. This is expressed as \(\lim_{x \rightarrow 2^+}(5x - 7)\).

Step 5: Create the Table for Right-hand Limit

Create a table with \(x\) values slightly greater than 2, like 2.1, 2.01, and 2.001. Compute their function values:- For \(x = 2.1\), \(f(2.1) = 5(2.1) - 7 = 3.5\)- For \(x = 2.01\), \(f(2.01) = 5(2.01) - 7 = 3.05\)- For \(x = 2.001\), \(f(2.001) = 5(2.001) - 7 = 3.005\)

Step 6: Observe the Limit from the Right-hand Side

As \(x\) approaches 2 from the right, the function values 3.5, 3.05, and 3.005 indicate that \(\lim_{x \rightarrow 2^+}(5x - 7)\) is approaching 3.

Step 7: Determine the Overall Limit

Since both left-hand and right-hand limits exist and are equal at 3, the overall limit is \(\lim_{x \rightarrow 2}(5x - 7) = 3\). This means the function approaches 3 as \(x\) approaches 2 from either direction.

Key Concepts

Left-Hand Limit

A left-hand limit of a function occurs when we examine the behavior of the function as the input variable approaches a specific value from values less than that specified number. This means observing what happens as we "sneak up" from the left side of the number line. In mathematical notation, this is expressed with a minus sign in the exponent, like this: \( \lim_{x \rightarrow a^-} f(x) \).To practically find this limit, let's take an example like \( \lim_{x \rightarrow 2^-}(5x - 7) \). We need to choose \( x \)-values that are slightly less than 2, such as 1.9, 1.99, and 1.999. These values are close to 2 but not equal to 2. For each of these \( x \)-values, compute the function's output:

• For \( x = 1.9 \), \( f(1.9) = 5 \times 1.9 - 7 = 2.5 \)
• For \( x = 1.99 \), \( f(1.99) = 5 \times 1.99 - 7 = 2.95 \)
• For \( x = 1.999 \), \( f(1.999) = 5 \times 1.999 - 7 = 2.995 \)

By observing these results, we can see that as \( x \) gets closer to 2 from the left, \( f(x) \) gets closer to 3. Thus, the left-hand limit is 3.

Right-Hand Limit

The right-hand limit is quite similar to the left-hand limit but focuses on approaching the value from the right. Essentially, we are checking what happens to the function as we get close to a number from slightly larger values.We write this limit using a plus sign in the exponent: \( \lim_{x \rightarrow a^+} f(x) \).Consider \( \lim_{x \rightarrow 2^+}(5x - 7) \). For this, we pick \( x \)-values slightly greater than 2, like 2.1, 2.01, and 2.001. Calculate the corresponding function values:

• For \( x = 2.1 \), \( f(2.1) = 5 \times 2.1 - 7 = 3.5 \)
• For \( x = 2.01 \), \( f(2.01) = 5 \times 2.01 - 7 = 3.05 \)
• For \( x = 2.001 \), \( f(2.001) = 5 \times 2.001 - 7 = 3.005 \)

Here, as \( x \) approaches 2 from the right, \( f(x) \) approaches 3. Consequently, the right-hand limit is 3.

Limit Calculation Steps

To successfully calculate limits, especially using the left-hand and right-hand approaches, follow these clear steps:

• Identify the point of interest: Determine which point you want the limit for and whether you are observing from the left or right.
• Select nearby values: Choose \( x \)-values close to the point you're interested in — less than the point for left limit and greater than it for the right limit.
• Compute function outputs: For each selected \( x \)-value, compute the function's output. This helps visually observe the trend.
• Observe the trend: Look for the value that the function seems to be approaching as \( x \) nears the point of interest.
• Determine if one or two-sided: If both limits (left and right) equal the same value, then that is the definitive limit at the point.

Always remember that the key to limits is understanding the approach rather than the exact value at the point, as limits discuss the trend towards a value.