Problem 19 - Calculus: Early Transcendentals

Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 4}(3 x-7)$$

Step-by-Step Solution

Answer

Answer: The limit is 5.

Step 1: Identify the limit

The limit we need to find is: $$\lim _{x \rightarrow 4}(3 x-7)$$

Step 2: Plug in the value of x

Since the function is continuous and well-defined, we can simply substitute the value of x in the limit: $$\lim _{x \rightarrow 4}(3 x-7)=3(4)-7$$

Step 3: Simplify

Now, we can simplify the expression to find the limit: $$3(4)-7=12-7=5$$ So, the limit is: $$\lim _{x \rightarrow 4}(3 x-7)=5$$

Key Concepts

Continuous Functions

When we talk about continuous functions in calculus, we refer to functions that are smooth and unbroken. This means there are no gaps or jumps in the graph of the function. A continuous function allows us to plug in the limit value directly to find the limit.
For a function to be continuous at a point, three conditions must be met:

• The function must be defined at that point.
• The limit of the function as it approaches that point must exist.
• The limit of the function as it approaches that point must equal the function's value at that point.

This straightforward process of substitution is possible only with continuous functions. In the given exercise, we used this property because the function \(3x-7\) is continuous everywhere on the real line.

Limit Evaluation

In calculus, evaluating limits helps us understand the behavior of functions as they approach a particular point. Evaluating a limit means finding the value that a function approaches as the input approaches a certain point.
In the step-by-step solution provided, we evaluated the limit \(\lim_{x \to 4}(3x - 7)\). Since the function is continuous, calculating the limit was straightforward by simply substituting \(x = 4\).
The steps for evaluating this limit were:

• Identify the expression and the point at which to evaluate the limit.
• Substitute the point into the expression.
• Simplify the result to find the limit, which in this case resulted in \(5\).

This substitution process highlights how powerful limits are for understanding sudden changes in function behavior.

Functions of Real Numbers

A function of real numbers is a rule that assigns each input from the realm of real numbers to exactly one output. These functions can have different forms, such as linear, polynomial, rational, and more.
The exercise involves the linear function \(3x-7\), which is an example of a simple function of real numbers. This function takes any real number, multiplies it by 3, and then subtracts 7.
Here’s a quick understanding of real number functions:

• They can portray relationships between varying quantities.
• They can model real-world phenomena like speed, temperature, or economics.
• Linear functions like \(3x-7\) are straightforward with a constant rate of change.

This concept is crucial in calculus as real number functions are often the basis for limit evaluation, continuity, and differentiability analyses.