Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(\lim _{x \rightarrow 2}[f(x)-f(2)] /(x-2)\) for each given function \(f\). $$f(x)=\frac{3}{x^{2}}$$

Step-by-Step Solution

Answer

The limit is \(-\frac{3}{4}\).

Step 1: Overview of the Problem

We are tasked with finding the limit \( \lim _{x \rightarrow 2}[f(x)-f(2)] /(x-2) \) for a function \( f(x) = \frac{3}{x^2} \). This is a classic problem of finding the derivative of a function at a point using the definition of the derivative.

Step 1: Calculate \( f(2) \)

First, calculate \( f(2) \) using the given function. Plug in \( x = 2 \) into the function:\[ f(2) = \frac{3}{2^2} = \frac{3}{4} \]

Step 2: Set up the Limit Expression

Now, set up the limit expression by substituting \( f(x) \) and \( f(2) \) into the formula:\[ \lim_{x \to 2} \frac{f(x) - f(2)}{x - 2} = \lim_{x \to 2} \frac{\frac{3}{x^2} - \frac{3}{4}}{x - 2} \]

Step 3: Simplify the Expression

Simplify the expression inside the limit. Find a common denominator for the fractions in the numerator:\[ \frac{3}{x^2} - \frac{3}{4} = \frac{12 - 3x^2}{4x^2} \]Now the limit becomes:\[ \lim_{x \to 2} \frac{12 - 3x^2}{4x^2(x - 2)} \]

Step 4: Factor and Cancel Terms

Factor the numerator \(12 - 3x^2\). It factors as:\[-3(x^2 - 4) = -3(x-2)(x+2)\].Substitute back into the limit:\[ \lim_{x \to 2} \frac{-3(x-2)(x+2)}{4x^2(x-2)} \]Cancel the \((x-2)\) terms:\[ \lim_{x \to 2} \frac{-3(x+2)}{4x^2} \]

Step 5: Evaluate the Limit

Now, substitute \( x = 2 \) into the simplified expression:\[ \frac{-3(2+2)}{4(2)^2} = \frac{-12}{16} = -\frac{3}{4} \]Thus, the limit is \(-\frac{3}{4}\).

Key Concepts

Limits

Limits are a foundational concept in calculus, helping us understand how functions behave as they approach a certain point. When we talk about the limit of a function as it approaches a particular value, we're examining the value that the function's outputs are getting closer to, as the input approaches the given value.
For example, in the problem, we are finding the limit of the expression \( \lim_{x \to 2}\frac{f(x) - f(2)}{x - 2} \). This examines how the function \( f(x) = \frac{3}{x^2} \) behaves near \( x = 2 \). Limits like these are crucial in calculus because they allow us to calculate derivatives.
Understanding limits requires getting comfortable with the idea that we're interested in behavior "just around" a point rather than exactly at the point itself. This is why we can sometimes disregard what happens at the exact point, focusing instead on closeness and trends.

Function Analysis

Function analysis involves examining the behavior and characteristics of functions to better understand their properties. For our exercise, we are analyzing the function \( f(x) = \frac{3}{x^2} \). This means looking at how the function behaves as \( x \) approaches a specific value – in this case, \( x = 2 \).
To analyze this function, we break it into parts like calculating \( f(2) \) to understand its specific value at a point, finding common denominators to simplify expressions and factorizing polynomials to solve limits effectively. Each step helps build a complete view of how the function operates in the context of our problem.
Additionally, function analysis often includes:

• Examining asymptotes, which are lines the graph approaches but never touches.
• Considering intervals of increase or decrease.
• Investigating points of discontinuity, where a function might "break" or not exist.

These analyses provide deep insights into how functions behave under various conditions.

Calculus Problem Solving

Calculus problem solving entails employing a structured approach to solve mathematical problems involving calculus principles, like derivatives and integrals. In our exercise, understanding the limit \( \lim_{x \to 2}[f(x)-f(2)]/(x-2) \) translates to finding the derivative of \( f(x) \) at a point – here specifically using the limit definition of the derivative.
The steps include:

• Determining the function values at the specified point to replace into the limit formula.
• Simplifying the equation, often necessitating algebraic manipulation such as factoring and canceling terms, as seen when we canceled \((x-2)\).
• Finally, evaluating the limit once it's in its simplest form.

This kind of step-by-step method is essential in calculus, ensuring no detail is missed, and each part of the function is thoroughly examined. Practice and familiarity with different types of functions and their behaviors enhance the ability to solve calculus problems efficiently.