Problem 53 - Precalculus with Limits

Question

In Exercises 49-68, find the limit by direct substitution. $$ \lim_{x \to 3}\ (-\dfrac{9}{x})$$

Step-by-Step Solution

Answer

The limit as \(x\) approaches 3 of \(-\dfrac{9}{x}\) is \(-3\).

Step 1: Identify the function and the point

The function is \(-\dfrac{9}{x}\) and the point that \(x\) is approaching is 3.

Step 2: Apply Direct Substitution

Replace \(x\) in the function with the value it's approaching. So, \(-\dfrac{9}{x}\) becomes \(-\dfrac{9}{3}\).

Step 3: Simplify the expression

Simplify \(-\dfrac{9}{3}\) which is \(-3\)

Key Concepts

Direct Substitution

Direct substitution is a technique used to find the limit of a function. It's one of the simplest methods for evaluating limits and works well for many functions without requiring complex algebra.
When we talk about direct substitution, we mean that you replace each occurrence of the variable in the function with the number or value that the variable is approaching.

• Identify the point that the variable is approaching.
• Substitute this point directly into the function.
• Simplify as needed to find the result.

For instance, if you're given the function \(-\dfrac{9}{x}\) and you need to find the limit as \(x\) approaches 3, you would simply substitute 3 for \(x\) and simplify to get \(-\dfrac{9}{3} = -3\).
This method is quick and effective when the function is continuous and defined for the value you're plugging in. However, be cautious with cases where direct substitution might lead to undefined expressions, such as division by zero.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In simple terms, it looks like a fraction where both the numerator and the denominator are polynomials.
The general form of a rational function is \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\).

• Numerator and denominator can have any degree.
• The function is undefined wherever the denominator is zero.

For example, the function \(-\dfrac{9}{x}\) is a rational function because it can be rewritten as \(\frac{-9}{x}\), where \(-9\) is the numerator and \(x\) is the denominator.
Understanding rational functions is important when evaluating limits because if a rational function has a point where its denominator is zero, it will be undefined there. In such cases, special techniques beyond direct substitution might be necessary.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus. It helps us understand the behavior of functions as they approach specific points.
When evaluating limits, we consider what value the function is getting closer to as the input approaches a particular point.

• One common method is direct substitution, which is used when the function is continuous at the point.
• If the function is not continuous or creates an indeterminate form, other strategies like factoring, rationalizing, or using L'Hôpital's rule might be necessary.
• Limits can help find the points where rational functions are undefined, revealing important characteristics about their graphs.

For the limit \(\lim_{x \to 3} (-\dfrac{9}{x})\), direct substitution works because \(-\dfrac{9}{x}\) is continuous at \(x = 3\), allowing us to simply substitute 3 for \(x\) and find that the limit is \(-3\).
Understanding limits provides insight into the continuity and asymptotic behavior of functions, making it a powerful tool in mathematical analysis.