Problem 79 - Thomas Calculus

Question

a. If $$\lim _{x \rightarrow 2} \frac{f(x)-5}{x-2}=3,\( find\ \)\lim _{x \rightarrow 2} f(x)$$ b. If $$\lim _{x \rightarrow 2} \frac{f(x)-5}{x-2}=4,\( find\ \)\lim _{x \rightarrow 2} f(x)$$

Step-by-Step Solution

Answer

Both parts: \( \lim_{x \to 2} f(x) = 5 \).

Step 1: Understand the given limit expression

We are given the limit expression \( \lim _{x \rightarrow 2} \frac{f(x)-5}{x-2}=3 \). This tells us that the expression \( \frac{f(x)-5}{x-2} \) approaches 3 as \( x \) approaches 2. This expression resembles the derivative of \( f(x) \) but is specifically formulated to find the immediate value of \( f(x) \) at \( x = 2 \).

Step 2: Relate the expression to the derivative

The expression \( \frac{f(x)-5}{x-2} \) can be seen as a form of the derivative \( \frac{f(x) - L}{x - a} \) where \( L \) is the limit of \( f(x) \) as \( x \) approaches \( a \). Here, we have \( L = 5 \) and \( a = 2 \). The expression approaching 3 suggests the slope of the tangent line at the point \( x = 2 \) is 3.

Step 3: Deduce \( \lim_{x \to 2} f(x) \) from the expression

Given \( \lim _{x \rightarrow 2} \frac{f(x) - 5}{x-2} = 3 \), we can interpret this as indicating that the limit of \( f(x) \) at \( x = 2 \) is 5. Essentially, the numerator \( f(x) - 5 \) must approach 0 as \( x \) approaches 2 to satisfy the limit equation.

Step 4: Apply the result to the modified limit

For the second problem, we have \( \lim _{x \rightarrow 2} \frac{f(x) - 5}{x-2} = 4 \). Despite the change in the slope indicator, this does not affect the value to which \( f(x) \) approaches as \( x \) approaches 2. Therefore, \( \lim_{x \to 2} f(x) \) remains 5 regardless of how the slope changes from 3 to 4.

Key Concepts

Limit Definition

One of the foundational concepts in calculus is the limit. In the context of this exercise, the limit examines how the expression \( \frac{f(x)-5}{x-2} \) behaves as \( x \) gets closer to 2. It's a way to understand the behavior of functions as they approach a specific point. The limit helps us predict the function's value at points where it might not be explicitly defined.

• The expression \( \lim _{x \rightarrow 2} \frac{f(x)-5}{x-2}=3 \) tells us that the expression is approaching 3 as \( x \) approaches 2.
• Using limits, we can infer that even if a function is not defined at a certain point, its behavior near the point can be understood.

The limit is a crucial tool for solving calculus problems, particularly when function behavior isn't straightforwardly available at a given x-value.

Derivatives

Derivatives provide a way to determine the rate of change of a function. They are fundamentally linked to the concept of limits, as they are defined as the limit of the difference quotient:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

• In this exercise, the expression \( \frac{f(x)-5}{x-2} \) closely resembles the concept of a derivative.
• The similarity implies that the slope of the tangent at the point \( x = 2 \) can be derived from the limit given in the problem.

Understanding derivatives is essential for examining how functions change and for problems involving tangent lines and rates of change.

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it. The notion of the tangent line at a point ties closely with the derivative at that point.

• The slope of the tangent line to the curve \( f(x) \) at a particular point \( x = a \) is given by the derivative, \( f'(a) \).
• In the first part of the exercise, the expression indicates that the slope of the tangent line at \( x = 2 \) is 3.

The tangent line offers insights into the instantaneous rate of change of a curve, providing a linear approximation at that point.

Continuity

Continuity of a function at a point ensures that the function behaves predictably and that there are no sudden jumps or holes at that point. For a function \( f(x) \) to be continuous at \( x = a \), the following must hold:

• \( \lim_{x \to a} f(x) = f(a) \)
• The function must be defined at \( x = a \).

The problem highlights that despite changes in the calculated limit (from 3 to 4), the continuity condition ensures that \( \lim_{x \to 2} f(x) = 5 \). This suggests that while the rate or slope might change, the function's overall behavior at the point does not.

Calculus Problem Solving

Solving calculus problems often involves breaking the problem down into smaller parts, understanding the underlying concepts, and applying them step-by-step. Let's look at how to approach limit problems effectively:

• Carefully analyze the given expressions and relate them to known calculus concepts like limits and derivatives.
• Understand the problem context—determine whether it involves finding a value, a rate of change, or understanding continuity.
• Apply the appropriate formulas and theorems, such as the limit definition theorem, to deduce the desired results.

In this exercise, recognizing the resemblance between the limit expression and the derivative, and understanding the implications for continuity, allowed us to solve for \( \lim_{x \to 2} f(x) \) effectively.