Problem 9 - Calculus Early Transcendentals

Question

If \(f\) and \(g\) are continuous functions with \(f(3)=5\) and $$\lim _{x \rightarrow 3}[2 f(x)-g(x)]=4,$$ find \(g(3)\)

Step-by-Step Solution

Answer

\( g(3) = 6 \)

Step 1: Identify Given Information

We are given two continuous functions, \( f \) and \( g \), with \( f(3) = 5 \) and the limit \( \lim_{x \rightarrow 3} [2f(x) - g(x)] = 4 \). We need to find \( g(3) \).

Step 2: Understanding Continuity

Because \( f \) and \( g \) are continuous functions, the values of \( f(x) \) and \( g(x) \) at \( x = 3 \) are equal to their respective limits as \( x \to 3 \). Therefore, \( \lim_{x \rightarrow 3} f(x) = f(3) = 5 \) and \( \lim_{x \rightarrow 3} g(x) = g(3) \).

Step 3: Substitute into Limit Expression

Substitute \( \lim_{x \rightarrow 3} f(x) = 5 \) into the limit expression: \( \lim_{x \rightarrow 3} [2f(x) - g(x)] = 4 \). This becomes \( \lim_{x \rightarrow 3} [2 \cdot 5 - g(x)] = 4 \).

Step 4: Simplify and Solve for \(g(3)\)

Simplify the equation \( \lim_{x \rightarrow 3} [10 - g(x)] = 4 \). Given \( g \) is continuous, this implies \( 10 - g(3) = 4 \), thus solving for \( g(3) \) gives \( g(3) = 6 \).

Key Concepts

Limit of a Function

In mathematics, the limit of a function describes the behavior of that function as its argument approaches a particular point. The idea is to understand what value a function is approaching, even if it's not defined at that specific point. To represent a limit, we use the notation \( \lim_{x \to a} f(x) = L \), where \( a \) is the point we are approaching, and \( L \) is the limit value that the function approaches.

Key aspects of limits include:

• The limit exists if and only if the function approaches the same value from both sides of the point \( a \).
• If a function is plotted, the spot where the function approaches a consistent value as \( x \) gets closer to \( a \) is its limit.

The exercise involves a limit expression: \( \lim_{x \rightarrow 3} [2f(x) - g(x)] = 4 \). This implies that as \( x \) nears 3, the value of the expression \( 2f(x) - g(x) \) approaches 4.

Understanding limits helps in finding points of continuity and makes it easier to deal with non-continuous parts of functions, which is vital for calculus applications.

Continuity

Continuity is a property of a function that ensures it behaves nicely without any breaks, jumps, or holes in its graph. A function is continuous at a point if the limit as you approach that point equals the function's value at that point. In more technical terms, a function \( f \) is continuous at \( a \) if \( \lim_{x \to a} f(x) = f(a) \).

Why is continuity important?

• It ensures that the function has no sudden changes in value as \( x \) changes.
• It provides smooth and predictable outcomes, making functions easier to analyze and manipulate.

In our exercise, both functions \( f \) and \( g \) are stated to be continuous. This is crucial because it allows us to equate the limit of these functions to their function values directly. Thus, \( \lim_{x \rightarrow 3} f(x) = f(3) \) and \( \lim_{x \rightarrow 3} g(x) = g(3) \). This property simplifies solving limits since the function behaves predictably around \( x = 3 \).

Solving Limits

Solving limits involves finding the value that a function approaches as its input approaches a certain point. It requires understanding the components of the function and recognizing the patterns in the behavior of these components. Here's how you can approach solving limits:

• First, substitute the value to see if the function is easily calculable and well-defined at that point.
• If direct substitution results in an indeterminate form, such as 0/0, you might need algebraic manipulation such as factoring or using conjugates.
• Apply limit laws and theorems, such as the limit of a sum is the sum of limits, to simplify expressions.

In the exercise, we solve the limit \( \lim_{x \rightarrow 3} [2f(x) - g(x)] = 4 \) by substituting the known value \( \lim_{x \rightarrow 3} f(x) = 5 \), making the expression \( \lim_{x \rightarrow 3} [10 - g(x)] = 4 \). Simplifying this gives us \( g(3) = 6 \).

This showcases how understanding limit properties helps solve complex equations and find unknown values in expressions involving continuous functions.

Function Values

Function values refer to the outputs of a function based on its inputs. For a function \( f(x) \), when we plug in a specific value for \( x \), we obtain a corresponding \( y \) value, \( f(x) \).

Key points about function values:

• They are determined by the rule or formula that defines the function.
• The function value at a particular point gives insight into the behavior of the function at that point.

In our original exercise, we have \( f(3) = 5 \). This means if we input 3 into the function \( f \), the result or output is 5. Similarly, through continuity, we found \( g(3) = 6 \).

These specific values are critical in solving limits and understanding the continuity of a function because knowing them can significantly simplify complex algebraic expressions during analysis.