Problem 11 - Calculus Early Transcendentals

Question

Evaluate the following limits. \(\lim _{x \rightarrow 4}(3 x-7)\)

Step-by-Step Solution

Answer

Answer: The limit of the function (3x-7) as x approaches 4 is 5.

Step 1: Identify the function

The given function to evaluate the limit is \((3x - 7)\).

Step 2: Identify the value at which the limit should be evaluated

We are asked to find the limit as \(x\) approaches \(4\), that is, \(\lim_{x\rightarrow 4}(3x-7)\).

Step 3: Substitute the value of \(x\)

Since the function is a linear function (polynomial of degree 1), it is continuous everywhere, including at \(x=4\). Therefore, we can directly substitute the value \(x=4\) to find the limit: $$\lim_{x\rightarrow 4}(3x-7) = 3(4) - 7$$

Step 4: Simplify the expression

Now, we will simplify the expression: $$3(4) - 7 = 12 - 7 = 5$$

Step 5: State the conclusion

The limit of the function \((3x-7)\) as \(x\) approaches \(4\) is \(5\). Therefore, $$\lim_{x\rightarrow 4}(3x-7) = 5$$

Key Concepts

Linear Functions

Linear functions are mathematical expressions that create straight lines when graphed. They have the general form \( f(x) = mx + b \), where \(m\) represents the slope, or rate of change, and \(b\) is the y-intercept, the point where the line crosses the y-axis. In our exercise, the function is \(3x - 7\), which is a linear function with a slope of 3 and a y-intercept of -7.

What makes linear functions special is their simplicity and predictability. Since they graph as straight lines, you can calculate any point on the line using the formula, without unexpected bends or curves.

When evaluating limits for a linear function, as in our example, the function's continuity simplifies the operation. Understanding that linear functions maintain constant rates of change helps reassure us that direct substitution is a reliable method for solving limits across the function's domain.

Continuous Functions

Continuous functions are functions that have no breaks, gaps, or jumps, meaning you can draw them without lifting your pencil from the paper. This attribute makes them predictable due to their unbroken flow from one point to another.

In the context of limit evaluation, continuous functions play a vital role because at any point \(c\), the limit of \(f(x)\) as \(x\) approaches \(c\) is simply \(f(c)\). This means you can substitute \(x = c\) directly into the function to find the limit.

• The function \(3x - 7\) in our exercise is continuous.
• This allows us to substitute \(x = 4\) without concern for discontinuity.
• We calculate \(3(4) - 7\) to easily find the limit.

Recognizing the continuity of a function simplifies limit problems significantly, eliminating the need for complex approaches in many scenarios.

Polynomial

Polynomials are expressions made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A typical polynomial of degree \(n\) has the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\).

In our exercise, \(3x - 7\) is a polynomial of degree 1 (also a linear function), simplifying the evaluation of limits.

• The degree of the polynomial is important as it indicates the highest power of the variable.
• Lower-degree polynomials, especially linear ones, are easier to work with because their graphs are simple straight lines or simple curves.
• Such polynomials are inherently continuous, making them predictable for limit evaluations.

Being familiar with polynomials helps us quickly identify and apply the right strategies for evaluating limits and recognizing continuity, which is handy for solving problems efficiently.