Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

, find \(\lim _{x \rightarrow 2}[f(x)-f(2)] /(x-2)\) for each given function \(\bar{f}\) $$ f(x)=\frac{1}{x} $$

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{4}\).

Step 1: Identify the Problem

We are asked to find the limit \(\lim_{x \rightarrow 2}\frac{f(x)-f(2)}{x-2}\) for the function \(f(x) = \frac{1}{x}\). This type of limit is related to the definition of a derivative, which requires understanding the function's behavior as \(x\) approaches a specific point.

Step 2: Determine the Value of the Function at the Point

First, we need to find \(f(2)\) by substituting \(x = 2\) into the function. Thus, \(f(2) = \frac{1}{2}\).

Step 3: Substitute Function and Point Values into Limit Expression

The expression inside the limit becomes \(\frac{f(x)-f(2)}{x-2} = \frac{\frac{1}{x} - \frac{1}{2}}{x-2}\). Simplifying this fraction is key to finding the limit.

Step 4: Find a Common Denominator

To simplify \(\frac{1}{x} - \frac{1}{2}\), find a common denominator, which is \(2x\). Rewrite as \(\frac{2-x}{2x}\).

Step 5: Cancel out the (x-2) Factor

Now, we have \(\frac{2-x}{2x(x-2)}\). Notice that \(2-x = -(x-2)\). Substitute this to get \(-\frac{(x-2)}{2x(x-2)} = -\frac{1}{2x}\). The \((x-2)\) terms cancel out.

Step 6: Evaluate the Limit

Now, as \(x \rightarrow 2\), the expression simplifies to \(-\frac{1}{2x}\). Evaluate it at \(x = 2\) to find \(-\frac{1}{2(2)} = -\frac{1}{4}\).

Key Concepts

Derivative Definition

Understanding the definition of a derivative is key to working with limits that resemble this expression: \(\lim_{x \rightarrow a}\frac{f(x) - f(a)}{x - a}\). This expression is closely related to the concept of the derivative of a function \(f(x)\) at a point \(x = a\). The derivative, \(f'(a)\), represents the instantaneous rate of change of the function at \(x = a\). It is formally defined as:

• \(f'(a) = \lim_{h \rightarrow 0}\frac{f(a+h) - f(a)}{h}\)

In the problem at hand, we are essentially finding the derivative of \(f(x) = \frac{1}{x}\) at \(x = 2\) using the alternative form of the derivative. By computing this limit, you are determining how the function \(f(x)\) behaves and changes precisely at \(x = 2\). This shows how derivatives allow us to analyze small changes and understand the behavior of functions at specific points.

Rational Functions

Rational functions are functions represented as the ratio of two polynomials. They are usually written in the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x) eq 0\).For the function \(f(x) = \frac{1}{x}\), it is a special type of rational function where the numerator is simply a constant, \(1\). Such rational functions can exhibit more complex behaviors. This includes asymptotic behavior or undefined points, depending on the polynomial in the denominator.Understanding rational functions helps solve limits like in this problem by knowing how to manipulate expressions and handle scenarios where the function may become undefined or approach an asymptote.

Limit Evaluation

Evaluating limits of functions is a crucial skill in calculus. When calculating a limit, especially as part of derivatives, one evaluates the behavior of a function as the input approaches a specific value.In our example, \(\lim_{x \rightarrow 2}\frac{f(x)-f(2)}{x-2}\), the challenge lies in the form \( \frac{0}{0} \), known as an indeterminate form. Solving such limits often involves algebraic manipulation. Here, you need to simplify the expression to find the limit, which resolves the indeterminacy, allowing you to evaluate the limit directly. By simplifying \(\frac{1}{x} - \frac{1}{2}\) to \(\frac{2-x}{2x}\), and recognizing \(2-x = -(x-2)\), the complexity of the limit evaluation becomes more manageable. This step is crucial for solving limits related to derivative definitions.

Function Simplification

Function simplification is essential when working with rational expressions, especially in calculus. Simplification can make it easier to evaluate expressions and solve limits.In our problem, simplifying \(\frac{\frac{1}{x} - \frac{1}{2}}{x-2}\) involved finding a common denominator to rewrite the subtraction in the numerator. This simplification step allowed us to cancel terms effectively and resolve the expression to \(-\frac{1}{2x}\). Key simplification steps include:

• Identifying common denominators for addition/subtraction within fractions.
• Recognizing equivalent expressions to simplify terms (like \(2-x = -(x-2)\)).
• Cancelling terms after factoring to remove complexities.

Mastering these simplification techniques is invaluable for limit evaluation, especially in derivative contexts.