Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated limit. \(\lim _{x \rightarrow 3} x\)

Step-by-Step Solution

Answer

The limit is \(\lim_{x \rightarrow 3} x = 3\).

Step 1: Identify the given function and value of x

The given function is f(x) = x, and we need to find the limit as x approaches 3.

Step 2: Check for continuity at the point x=3

The function f(x) = x is continuous and defined at x=3. Due to the continuity, we can directly substitute the value of x into the function.

Step 3: Substitute the value of x into the function

Now, we can substitute the value of x = 3 into the function f(x) = x: \(f(3) = 3\).

Step 4: Determine the limit

Since the function is continuous at x=3 and we have found the value of the function at x=3, the limit is: \(\lim_{x \rightarrow 3} x = 3\).

Key Concepts

Continuity

Continuity is an essential concept in calculus that helps us understand how functions behave. A function is said to be continuous at a point if there is no "break" or "jump" at that point. Imagine drawing the graph of a function with a pencil without lifting it from the paper; this indicates continuity.
In mathematical terms, a function \( f(x) \) is continuous at a point \( x = a \) if:

• The function \( f(x) \) is defined at \( x = a \).
• The limit \( \lim_{x \to a} f(x) \) exists.
• The limit \( \lim_{x \to a} f(x) \) equals the function value \( f(a) \).

For the function \( f(x) = x \), it's continuous at every value of \( x \), including \( x = 3 \). This continuity implies that we can find the limit by directly evaluating the function at that point.

Substitution Method

The substitution method is a straightforward approach used to find limits, especially when dealing with continuous functions. When a function is continuous at a certain point, you can substitute the value directly into the function to determine the limit.
This simplicity is particularly handy as it saves from complications that might arise in more complex functions. For a function \( f(x) = x \), as we approach \( x = 3 \), we simply substitute \( x = 3 \) into \( f(x) \) to get \( f(3) = 3 \).
This method works best with polynomials and other elementary functions where continuity is straightforward. If the function is not continuous at a point, then we might need other methods like factoring or rationalization.

Limit of a Function

The concept of a limit is one of the fundamental building blocks of calculus. It describes the behavior of a function as it approaches a specific point. In simpler terms, the limit defines what value a function "approaches" as the variable gets closer to a certain number.
To find the limit of a function \( \lim_{x \rightarrow a} f(x) \), you're essentially asking "What does \( f(x) \) get closer to as \( x \) approaches \( a \)?".
In our case, with \( f(x) = x \), as \( x \) approaches 3, the function itself naturally approaches 3. This intuitive behavior demonstrates how the concept of limits is closely linked with continuity. When a function is continuous and simple, the limit often mirrors the function value at that point, such as \( \lim_{x \rightarrow 3} x = 3 \).
Understanding limits helps in analyzing functions' behaviors at their boundaries and ensures smoother transitions across values.