Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

, find each of the right-hand and left-hand limits or state that they do not exist. $$ \lim _{x \rightarrow 3^{+}}\left[x^{2}+2 x\right] $$

Step-by-Step Solution

Answer

The right-hand limit is 15.

Step 1: Understand Right-Hand Limit Definition

The right-hand limit of a function as \(x\) approaches a certain value \(c\) is denoted by \(\lim_{{x \to c^+}} f(x)\). It calculates the behavior of \(f(x)\) as \(x\) approaches \(c\) from the right (i.e., with values greater than \(c\)).

Step 2: Identify the Function

The function provided is \(f(x) = x^2 + 2x\). We need to find the behavior of this function as \(x\) approaches 3 from the right.

Step 3: Substitute the Approach Value

For right-hand limits, directly substitute the value of \(x\) we are approaching (from the right) into the function.\[\lim_{{x \to 3^+}} \left[x^2 + 2x\right] = 3^2 + 2 \times 3.\]

Step 4: Calculate the Limit

Evaluate the substituted expression:\[3^2 + 2 \times 3 = 9 + 6 = 15.\] Therefore, \(\lim_{{x \to 3^+}} \left[x^2 + 2x\right] = 15.\)

Key Concepts

Right-hand limit

The right-hand limit is a concept used in calculus to understand the behavior of functions as the variable approaches a specific point from the right side. Imagine you're walking on a number line and want to understand how close you can get to a certain number from the high side. That's what a right-hand limit helps you determine.

In mathematical terms, if you're finding the right-hand limit as \(x\) approaches a number \(c\), you use the notation \(\lim_{{x \to c^+}} f(x)\). This expression essentially means you are approaching \(c\) with values slightly larger than \(c\). It is an excellent way to predict and comprehend the function's behavior right before it actually gets to \(c\).

• The function in this case is \(f(x) = x^2 + 2x\).
• We evaluate it for \(x\) approaching 3 from the right, expressed as \(\lim_{{x \to 3^+}} (x^2 + 2x)\).

Limit evaluation

Limit evaluation is the process of determining the number a function approaches as the input approaches a particular value. This is a core tool in calculus, useful for investigating the behavior of functions at points that sometimes aren't easy to compute directly.

When evaluating limits, consider the following steps:

• Identify the approach value, which is 3 in our case.
• Substitute this value into the function to see if you can directly compute the limit.
• Ensure the result makes sense by reconsidering the function’s behavior around the approach value.

Once all these steps are completed, you understand the function better. The process essentially lets you peer into the function's future behavior as it gets closer to a point.

Substitution method

The substitution method is a straightforward technique used for evaluating limits. It's almost like testing how a function behaves at specific points by inputting those values directly.

To use this method, simply substitute the value you are approaching into the function expression:

• In this example, the function is \(f(x) = x^2 + 2x\).
• Since we're finding \(\lim_{{x \to 3^+}} (x^2 + 2x)\), we plug \(x = 3\) into the formula.
• This leads us to calculate \(3^2 + 2 \times 3 = 9 + 6 = 15\).

The approach nicely calculates the value of the function as \(x\) gets closer and closer to 3 from the right. It provides a direct route to obtain limit values and is usually the first step in easier limit problems. However, remember this method isn't always suitable, particularly when direct substitution results in undefined forms.