Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→4(3x−7)

Verified step by step guidance

1. Identify the type of limit problem: This is a basic limit problem where you need to find the limit of a linear function as x approaches a specific value.
2. Recognize that the function given is a linear polynomial, \(3x - 7\), which is continuous everywhere.
3. For continuous functions, the limit as x approaches a value is simply the value of the function at that point.
4. Substitute the value that x is approaching into the function: Replace x with 4 in the expression \(3x - 7\).
5. Simplify the expression after substitution to find the limit.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for defining continuity and derivatives. In this case, we are interested in the limit of the function as x approaches 4.

Substitution Method

The substitution method is a technique used to evaluate limits by directly substituting the value that x approaches into the function, provided the function is continuous at that point. If the function is defined at that point, this method often yields the limit directly. For the given limit, substituting x = 4 into the function 3x - 7 will help find the limit.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This concept is essential when evaluating limits, as it ensures that the function behaves predictably around the point of interest. The function 3x - 7 is a polynomial, which is continuous everywhere, including at x = 4.