Problem 29 - 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. $$\lim _{x \rightarrow 3} \frac{-5 x}{\sqrt{4 x-3}}$$

Step-by-Step Solution

Answer

Answer: The limit of the function as x approaches 3 is -5.

Step 1: Identify the Function

The given function is: $$f(x) = \frac{-5x}{\sqrt{4x-3}}$$ The goal is to find the limit of this function as x approaches 3: $$\lim_{x \rightarrow 3} f(x)$$

Step 2: Direct Substitution

We start by attempting to plug in x = 3 directly into the function to see if we can determine the limit: $$f(3) = \frac{-5(3)}{\sqrt{4(3)-3}}$$ Simplifying the expression: $$f(3) = \frac{-15}{\sqrt{9}}$$ Since the denominator is not zero, we can proceed with the direct substitution:

Step 3: Simplifying the Expression

Simplify the expression to compute the limit: $$f(3) = \frac{-15}{\sqrt{9}} = \frac{-15}{3} = -5$$ So, the limit as x approaches 3 is: $$\lim_{x \rightarrow 3} \frac{-5x}{\sqrt{4x-3}} = -5$$

Key Concepts

Direct Substitution in Limits

Direct substitution is often the first method used to evaluate a limit. It involves plugging the value that x approaches into the function directly. This is the simplest technique and is applicable when the function is continuous at the point of interest.

For example, here we have:

• Function: \( f(x) = \frac{-5x}{\sqrt{4x - 3}} \)
• Point of interest: \( x = 3 \)

To use direct substitution:

• Replace \( x \) with 3 in the function.
• If the function evaluates to a finite number, the limit at that point exists.
• Should the substitution result in an indeterminate form, like \( \frac{0}{0} \), further analysis is needed.

In this specific exercise, substituting \( x = 3 \) results in a clear and finite answer, indicating the limit can be found directly by substitution.

Limit Evaluation

Evaluating a limit provides critical insights into the behavior of a function as its input approaches a particular value. It's essentially about understanding how functions behave near specific points.

To evaluate limits, one can generally consider:

• Direct substitution: If the function is continuous at the point, simply substitute the target value.
• Factorization: Simplify or cancel terms if direct substitution gives an indeterminate form.
• L'Hôpital's Rule: Useful when limits result in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

In the given problem, evaluations through direct substitution showed no problematic terms, leading to an immediate simplification yielding \( -5 \).

Real Numbers in Calculus

In calculus, real numbers are fundamental. They include all the rational and irrational numbers, serving as the backbone for defining limits, continuity, and derivatives. When evaluating limits, we often work within the realm of real numbers.

Key properties of real numbers used in limits include:

• Arithmetic operations: Addition, subtraction, multiplication, and division are well-defined unless division by zero occurs.
• Existence: Limits are said to exist when both the left-hand and right-hand limits approach the same real number.
• Approaching values: The concept of 'approaching' is inherently tied to real numbers, as we evaluate what happens as \( x \) gets infinitely close to a specific point.

The problem solved involves a real number limit evaluation, affirming the utility of the direct substitution when limits behave predictably under real number conditions.