Problem 31 - Calculus Early Transcendentals

Question

Evaluate the following limits. \(\lim _{x \rightarrow 3} \frac{-5 x}{\sqrt{4 x-3}}\)

Step-by-Step Solution

Answer

\(-5\)

Step 1: Identify the Limit

We evaluate:
Evaluate the following limits.
\(\lim _{x \rightarrow 3} \frac{-5 x}{\sqrt{4 x-3}}\)

Step 2: Attempt Direct Substitution

Substitute the approach value. If defined, that is the limit. If indeterminate (\(0/0\)), apply algebraic techniques or L'Hopital's Rule.

Step 3: Apply Limit Laws if Needed

Use sum, product, quotient, and power limit laws to justify the result.

Step 4: Result

The limit is: Question: Evaluate the limit: \(\lim _{x \rightarrow 3} \frac{-5 x}{\sqrt{4 x-3}}\) Answer: -5

Key Concepts

Limit Evaluation

Calculus limits are a fundamental concept used to understand the behavior of functions as they approach a certain point. Evaluating limits helps in analyzing the behavior of mathematical functions in various scenarios, particularly when functions do not behave nicely at a given point. For this exercise, we're interested in evaluating the limit as \(x\) approaches a specific number.

• The limit can tell us about the value that a function approaches as the input gets closer to a certain point.
• Understanding limits is essential for defining derivatives and integrals, which are core elements of calculus.
• One common situation is finding the limit of a rational function, which involves a fraction with polynomial expressions in the numerator and denominator.

In our example, the goal is to find the limit of the function \( \frac{-5x}{\sqrt{4x-3}} \) as \( x \) approaches 3. This involves analyzing how the function behaves and calculating its value precisely. If you can substitute directly without causing any undefined scenarios (like division by zero), then you're on the path to solving it easily.

Direct Substitution

Direct substitution is one of the simplest methods for evaluating a limit. It involves directly substituting the value that \( x \) approaches into the expression. This method is useful when the function is continuous and well-behaved at that point, meaning there are no breaks, jumps, or undefined situations.In cases of direct substitution:

• Simply plug the approaching value into the function.
• Check to see if the result is a valid number.
• If the result is defined, that is the limit.

In this exercise, direct substitution is successful since substituting \( x = 3 \) into the function \( \frac{-5x}{\sqrt{4x-3}} \) leads to:\(-5(3)\) over \(\sqrt{4(3)-3}\), simplifying to \(\frac{-15}{3}\) which equals \(-5\).The direct substitution yields a valid result without any complexity, confirming that the limit is \(-5\). This situation showcases the simplicity and effectiveness of using direct substitution when suitable.

Rational Functions

Rational functions are a type of function that involves ratios of polynomials. The expression \( \frac{-5x}{\sqrt{4x-3}} \) is an example, even though it includes a square root in the denominator, the structure is akin to that of a rational function.

• A rational function generally appears in the form \( \frac{p(x)}{q(x)} \) where both \( p(x) \) and \( q(x) \) are polynomials.
• When evaluating limits, particular care must be taken with these functions, due to potential divisions by zero or undefined points.
• Occasionally, algebraic manipulation or factoring is essential before applying other limit evaluation techniques.

However, in our expression, the direct substitution method worked effectively because plugging the value directly into \( x \) did not produce any undefined operation such as division by zero. It highlights how sometimes, solving limit problems can be straightforward when a rational function is well-behaved at the point of interest.