Problem 1 - Calculus of a Single Variable

Question

Estimating a Limit Numerically In Exercises 1–6, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 4} \frac{x-4}{x^{2}-3 x-4} $$

Step-by-Step Solution

Answer

The limit of the function as \(x\) approaches 4 is \( \frac{1}{5} \).

Step 1: Simplify The Expression

First, simplify the function by factoring the denominator.\n The function \( f(x) = \frac{x-4}{x^{2}-3x-4} \) can be simplified to \( f(x) = \frac{x-4}{(x-4)(x+1)} \). After canceling \( x-4 \) from both the numerator and the denominator, the function is simplified to \( f(x) = \frac{1}{x+1} \).

Step 2: Substitute the Limit into The Simplified Function

The limit of the function as \( x \) approaches 4 can be evaluated by substitifying \( x = 4 \) into the simplified function. \n So, \( \lim_{x \rightarrow 4} f(x) = f(4) = \frac{1}{4+1} = \frac{1}{5} \).

Step 3: Confirm Result Graphically with a Graphing Utility

A graph of the function \( f(x) = \frac{x-4}{x^{2}-3x-4} \) can be plotted using any graph plotting utility. The function approaches \( \frac{1}{5} \) as \( x \) approaches 4. This confirms the algebraic solution.

Key Concepts

Limits of Functions

Understanding the concept of limits is crucial in calculus. Limits describe the behavior of a function as the input approaches a certain value. They are foundational for defining continuity, derivatives, and integrals.
In our exercise, we explore the limit of a fraction where the numerator and denominator both become zero when evaluated at the limit point. This scenario is known as an indeterminate form. To resolve this, we apply factorization, simplifying the expression to avoid division by zero at the limit point. This allows us to estimate the limit numerically or to confirm it graphically using graphing tools.

Factorization in Calculus

Factorization is a powerful tool in calculus, used to simplify expressions and solve equations. It brings complex polynomials back to a product of simpler ones, often revealing solutions that are not readily visible. In the context of our exercise, we factor the denominator to reveal a common factor in both the numerator and denominator. This technique, also known as 'canceling out', is a common step when evaluating limits of functions with indeterminate forms. Proper factorization is essential for successfully estimating the limit of a function. Here's how it was done:

• We started with the original function \( f(x) = \frac{x-4}{x^2-3x-4} \).
• The denominator \( x^2-3x-4 \) was factored to \( (x-4)(x+1) \).
• The common factor \( x-4 \) in both numerator and denominator was canceled.
• This simplification reveals a clearer expression \( \frac{1}{x+1} \) to work with.

Graphing Utilities in Calculus

Graphing utilities are indispensable in calculus, allowing students and professionals to visualize functions and their limits. A graph can confirm what numbers and algebra suggest, providing intuitive understanding of a function's behavior as values approach a specific point.
In the exercise, after simplifying the function and estimating the limit numerically, we confirm our result graphically. By plotting \( f(x) = \frac{x-4}{x^2-3x-4} \) as \( x \) approaches 4, we expect to see the graph approach the value we calculated: \( \frac{1}{5} \). The visual confirmation through a graphing tool compliments our algebraic findings, and serves as a valuable cross-reference, ensuring a more thorough understanding of the limit concept.