Problem 17 - Contemporary Precalculus

Question

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 4^{-}} \frac{x-4}{x^{2}-16}$$

Step-by-Step Solution

Answer

Answer: The limit of the function as x approaches 4 from the left side is \(\frac{1}{8}\).

Step 1: Simplify the given function

First, we can simplify the given function by factoring the numerator and the denominator. $$ \frac{x-4}{x^{2}-16} = \frac{x - 4}{(x - 4)(x + 4)} $$

Step 2: Cancel out the common factors

Now we can cancel out the common factors in the numerator and the denominator. $$ \frac{x - 4}{(x - 4)(x + 4)} = \frac{span1}{(x + 4)} $$

Step 3: Substitute the limiting value

We have to find the limit as x approaches 4 from the left side. We can substitute the value of x approaching 4 from the left side (slightly smaller than 4) into the simplified function. $$ \lim _{x \rightarrow 4^{-}} \frac{1}{x + 4} = \frac{1}{4 + 4} $$

Step 4: Evaluate the limit

Finally, we evaluate the limit. $$ \frac{1}{4 + 4} = \frac{1}{8} $$ So, the limit of the function as x approaches 4 from the left side is \(\frac{1}{8}\).

Key Concepts

Limit Existence

Limits in calculus help us understand the behavior of a function as it approaches a particular point. One key question is: does the limit exist? For the limit to exist, the function must approach a specific value from both sides of the point. In the expression \( \lim_{x \rightarrow 4^{-}} \frac{x-4}{x^{2}-16} \), we focus on the left approach to 4.
This means we check behavior as \( x \) gets closer to 4 from values smaller than 4. The concept of limit existence ensures that we can predictably identify the value the function will approach, avoiding undefined operations such as division by zero.

Factoring in Mathematics

Factoring is a technique in mathematics used to simplify expressions by finding the common elements. In our expression \( \frac{x-4}{x^{2}-16} \), the denominator can be factored. The denominator \( x^2 - 16 \) is a difference of squares, which can be factored as \((x-4)(x+4)\).
By simplifying the expression through factoring, we can cancel the common factor \((x-4)\) in the numerator and denominator. This reduces the expression to \( \frac{1}{x+4} \).

• Factoring is often the first step in simplifying rational functions.
• It helps to avoid indeterminate forms, making it easier to solve limits.

Limit Evaluation

Once we've simplified the expression \( \frac{1}{x+4} \), evaluating the limit becomes straightforward. With the function reduced, substitute the limiting value into the simplified form. As \( x \to 4^- \), or as \( x \) approaches 4 from the left, it is crucial to substitute values just under 4.
After substitution, the function \( \frac{1}{x+4} \) becomes \( \frac{1}{4+4} \), which simplifies to \( \frac{1}{8} \).

• Ensure all indeterminate forms are resolved before substituting.
• This process confirms the limit value precisely.

Rational Functions

Rational functions are expressions of the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. Our given problem had a rational function \( \frac{x-4}{x^{2}-16} \). These functions often involve techniques like factoring to simplify.
By understanding rational functions, we learn how to identify limit behavior, especially around points where the denominator may become zero. This is key when \( x \to 4 \), which initially makes \( x^2-16 \) zero.

• Simplifying rational functions helps manage potential discontinuities.
• An important skill is recognizing when to factor as a tool for simplification.