Problem 10 - Calculus and its applications

Question

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 1}(3 x+2) $$

Step-by-Step Solution

Answer

The limit is 5.

Step 1: Understand the Function

The given function is a linear function, which is represented as \(f(x) = 3x + 2\). We are asked to find \(\lim_{x \rightarrow 1}(3x+2)\).

Step 2: Apply the Theorem on Limits of Rational Functions

The Theorem on Limits of Rational Functions states that for polynomial functions like linear functions, we can directly substitute the limit point into the function to find the limit. Linear functions are continuous everywhere, so limits exist wherever they are evaluated.

Step 3: Substitute the Limit Point into the Function

Substitute \(x = 1\) into the function \(3x + 2\). Calculate \(3(1) + 2 = 3 + 2 = 5\).

Step 4: Conclusion

Since the direct substitution results in a finite value, the limit exists and is equal to 5.

Key Concepts

Linear Function

A linear function is one of the simplest types of functions in mathematics. It is called "linear" because when the function is expressed as \(f(x) = ax + b\), where \(a\) and \(b\) are constants, it graphs as a straight line. The equation is characterized by:

• \(a\) represents the slope of the line, which indicates the steepness or flatness.
• \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.

For instance, in the function \(3x + 2\), \(3\) is the slope, and \(2\) is the y-intercept. Linear functions are straightforward to analyze because their constant slope means they have no curvature. This quality makes it easy to calculate limits since the slope is consistent across the function's domain.

Polynomial Functions

Polynomial functions are expressions that involve sums and products of constants and variables. They are of the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n\) is a constant. The power \(n\) is a non-negative integer, making polynomials a versatile class of functions.

• The degree of a polynomial is the largest power of \(x\) that appears with a non-zero coefficient.
• Each term in a polynomial is called a monomial, and polynomials can become increasingly complex with higher degrees.

The function \(3x + 2\) is a first-degree polynomial, also recognized as a linear polynomial. Such functions are especially easy to work with concerning limits and other mathematical operations due to their simplicity.

Theorem on Limits

The Theorem on Limits assures us that for rational and polynomial functions, including linear functions, the limit can be found by simply substituting the value to which \(x\) approaches directly into the function. This theorem provides ease in evaluating limits provided the function is continuous at the point.

• If the function is a polynomial or rational function without division by zero or other undefined operations, direct substitution is valid.
• Linear functions, being continuous everywhere, allow the application of this theorem without any conditions.

In our example of finding the limit of \(3x + 2\) as \(x\) approaches \(1\), by substituting directly, we get \(3(1) + 2 = 5\). Thereby, the theorem guarantees that this limit exists and is calculable directly.