Problem 6 - Calculus Single Variable

Question

In Exercises \(1-8\), evaluate the given limit. $$ \lim _{x \rightarrow 3} \frac{x^{2}-4 x+3}{x-3} $$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Substitution Check

First, substitute the value of 3 into the limit expression to check if it results in an indeterminate form or a numeric value. Substitute: \( \frac{3^2 - 4 \times 3 + 3}{3 - 3} \). This simplifies to \( \frac{9 - 12 + 3}{0} = \frac{0}{0} \), which is an indeterminate form. This confirms that we need to manipulate the expression further to find the limit.

Step 2: Factor the Numerator

To resolve the indeterminate form, factor the quadratic expression in the numerator. The expression \( x^2 - 4x + 3 \) can be factored as \( (x - 1)(x - 3) \). This gives us the new expression: \( \frac{(x - 1)(x - 3)}{x - 3} \).

Step 3: Simplify the Expression

Now simplify by canceling the \((x - 3)\) terms in the numerator and denominator, as long as \(x eq 3\). This leaves us with the simplified expression \( x - 1 \).

Step 4: Evaluate the Limit

Substitute \(x = 3\) into the simplified expression \(x - 1\). This yields \(3 - 1 = 2\). Thus, the limit is 2.

Key Concepts

Limit Evaluation

When we discuss limit evaluation in calculus, we are asking what value a function approaches as the input approaches a certain number. It's like trying to predict where a running train is heading once it reaches a bend on the tracks. In essence, knowing limits helps us understand the behavior of functions near specific points.
For instance, consider the limit provided in the exercise, \( \lim_{x \rightarrow 3} \frac{x^{2}-4x+3}{x-3} \). Initially, you might try simple substitution by plugging the number 3 directly into the equation. This step helps check whether the function returns a specific value or an undefined form like \( \frac{0}{0} \). In this example, inserting \( x = 3 \) directly leads to an indeterminate form, so further steps are necessary.
Evaluating limits is crucial when dealing with functions that are not easy to approach directly, ensuring we fully understand the function's behavior around a given point. It's like a detective solving a mystery, finding clues to discover how a function behaves as it nears our number of interest.

Factoring Quadratics

Factoring quadratics is an algebraic process useful in solving equations, especially when evaluating limits. When you have a function with a quadratic in the numerator, like \( x^2 - 4x + 3 \), factoring can simplify the expression to remove troublesome terms.
In the exercise, the quadratic \( x^2 - 4x + 3 \) is factored to \((x - 1)(x - 3)\). Factoring seeks numbers (or expressions) whose product equates to the quadratic, much like how you can split 6 into 2 and 3 because 2 multiplied by 3 gives 6.
This strategic breakdown allows you to simplify the fraction by canceling common terms in the numerator and denominator. After cancellation, we are left with a simpler expression that is easier to handle. This is especially helpful if initial substitution into the limit results in an indeterminate form like we saw before. The goal is always to clean up the expression so that it can be evaluated directly.

Indeterminate Forms

In calculus, encountering an indeterminate form signifies that an expression is initially undefined with standard arithmetic rules. The \( \frac{0}{0} \) form is a classic and common case that requires additional steps to resolve. It's similar to trying to answer 'zero divided by zero apples'—it doesn't make logical sense without more manipulation.
When the limit \( \lim_{x \rightarrow 3} \frac{x^{2}-4x+3}{x-3} \) produces the indeterminate form \( \frac{0}{0} \), further algebraic manipulation, like factoring, is necessary. This manipulation uncovers the true path of the limit.
Indeterminate forms signal hidden information within the mathematical expression demanding a deeper dive. By simplifying these forms, we can hone in on the actual behavior of the function, often revealing finite and meaningful outputs that weren't initially apparent. It’s like clearing the fog to see the path clearly.