Problem 22 - Calculus: Early Transcendentals

Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 6} 4$$

Step-by-Step Solution

Answer

Answer: The limit of the function 4 as x approaches 6 is 4.

Step 1: Identify the function

Since the function given is \(4\), which is a constant function, we can proceed to apply the property of limits for constant functions.

Step 2: Apply the Constant Function Limit Property

The limit of a constant function is simply the constant itself, regardless of the value that x is approaching. Therefore: $$\lim _{x \rightarrow 6} 4 = 4$$

Step 3: State the Answer

The limit of the function 4 as x approaches 6 is 4.

Key Concepts

Constant Function

A constant function is about as straightforward as it sounds. It's a function where the value doesn't change, no matter what input you provide. Imagine it like a flat line on a graph; everywhere you look, the height remains the same. In mathematical terms, a function is constant if it can be written as \( f(x) = c \), where \( c \) is a fixed real number and does not depend on \( x \). For example, if you were given \( f(x) = 4 \), no matter what \( x \) is, \( f(x) \) will always be 4.
This concept is quite straightforward, but crucial in understanding limits. Since the function doesn't vary with \( x \), the limit of a constant function as \( x \) approaches any number is the constant itself. So, \( \lim_{x \to a} c = c \) for any real number \( a \). This property helps simplify problems where you're asked to find limits of straightforward constant functions.

Limit Properties

When working with limits, several useful properties can simplify complex issues into more manageable parts. These properties let us break down what we're working with, making even the trickiest calculations easier.

• Constant Rule: As mentioned above, for any constant \( c \), \( \lim_{x \to a} c = c \).
• Sum or Difference Rule: The limit of a sum or difference is the sum or difference of the limits. Such as, \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \).
• Product Rule: The limit of a product is the product of the limits. This means, \( \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \).
• Quotient Rule: Likewise, the limit of quotient is the quotient of the limits, provided the denominator isn't zero: \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \), if \( \lim_{x \to a} g(x) eq 0 \).

Each of these rules lets you simplify an equation before applying any specific, complicated limits. They are the building blocks for handling any type of limit problem you encounter.

Real Numbers

Real numbers are the foundation of many mathematical disciplines. They are like the playground where calculus comes alive. These numbers encompass all decimal expansions, both rational and irrational. Rational numbers are those that can be expressed as fractions, like \( \frac{1}{2} \) or \( -7.2 \). Irrational numbers, such as \( \sqrt{2} \) or \( \pi \), cannot be written precisely as fractions, due to their non-repeating, non-terminating decimal nature. Real numbers are used all the time when talking about limits. You often see limits involving bounds that are real numbers, signaling that the variable \( x \) approaches some value within the set of real numbers. Real numbers ensure no holes or gaps within the number set, providing a complete set where limits are continually relevant. In the exercise, the constant \( 4 \) and the approaching value \( 6 \) are real numbers, showing that this concept is fundamental to understanding where your limits will land.