Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4}[f(x)-g(x)]$$

Step-by-Step Solution

Answer

The limit is 8.

Step 1: Identify Limit Rule

Recognize that the problem asks for the limit of the difference of two functions as \(x\) approaches 4, given individual limits. According to limit rules, the limit of a difference is the difference of the limits: \(\lim_{x \to a}(f(x) - g(x)) = \lim_{x \to a}f(x) - \lim_{x \to a}g(x)\).

Step 2: Substitute Known Limits

Substitute the given limits into the rule. You have \(\lim_{x \to 4}f(x) = 16\) and \(\lim_{x \to 4}g(x) = 8\). Therefore, substitute these values into the formula: \(\lim_{x \to 4}[f(x) - g(x)] = 16 - 8\).

Step 3: Calculate the Limit

Perform the subtraction to find the limit. Calculate \(16 - 8 = 8\). Thus, \(\lim_{x \to 4}[f(x) - g(x)] = 8\).

Key Concepts

Limit Rules

Limits are fundamental to calculus, representing the value a function approaches as the input reaches a certain point. The limit rules are handy tools that make working with limits simpler, especially when dealing with complex functions. One crucial rule to remember is that limits behave linearly. This means the limit of a sum is the sum of the limits, and similarly, the limit of a difference is the difference of the limits. These properties arise because limits are about approaching values, not fixed points. For example, if you have the limits of two functions, like \( \lim_{x \rightarrow a} f(x) \) and \( \lim_{x \rightarrow a} g(x) \), and you're asked to find the limit of their sum or difference, you simply add or subtract the individual limits. With these rules, we can solve limits in a straightforward manner by breaking down complex expressions into simpler parts.

Difference of Functions

When dealing with limits, a common scenario is finding the limit of a difference between two functions. This concept is applied when you have two functions, \(f(x)\) and \(g(x)\), and you need to evaluate \(\lim_{x \rightarrow a} [f(x) - g(x)]\). Thanks to the linearity of limits, this problem simplifies to just taking the limit of each function independently and then finding their difference.

• Step 1: Determine the limit of each function separately: \(\lim_{x \rightarrow a} f(x)\) and \(\lim_{x \rightarrow a} g(x)\).
• Step 2: Calculate the difference: \(\lim_{x \rightarrow a} f(x) - \lim_{x \rightarrow a} g(x)\).

This simple breakdown allows us to handle even complex functions by reducing them to manageable parts. This is especially useful in calculus, where functions can often be unwieldy and difficult to work with directly.

Substitution Method

The substitution method is a straightforward technique for solving limit problems when the limits of functions are already known. This method leverages the known limits of individual functions to find the limit of a composite expression. Here's how it works:- Identify the individual limits. For instance, if you are given \(\lim_{x \rightarrow a} f(x) = 16\) and \(\lim_{x \rightarrow a} g(x) = 8\), mark these values.- When tasked with finding a combined limit like \(\lim_{x \rightarrow a} [f(x) - g(x)]\), directly substitute the known limits into the expression.This direct substitution saves time and effort by allowing you to bypass more intricate methods of limit calculation. It assumes that the functions involved are sufficiently well-behaved around the point in question. By using the substitution method, what could potentially be a complex problem becomes a simple arithmetic operation, highlighting the power and elegance of understanding and utilizing limits in calculus.