Problem 15 - Brief Applied Calculus

Question

Find the following limits without using a graphing calculator or making tables. $$ \lim _{x \rightarrow 5} \frac{3 x^{2}-5 x}{7 x-10} $$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Check for Direct Substitution

Start by substituting the value of \( x = 5 \) directly into the function \( \frac{3x^2 - 5x}{7x - 10} \). Calculate \( 3(5)^2 - 5(5) \) and \( 7(5) - 10 \).\[3(5)^2 - 5(5) = 75 - 25 = 50\]\[7(5) - 10 = 35 - 10 = 25\]Thus, the direct substitution gives you \( \frac{50}{25} = 2 \) if the denominator is not zero.

Step 2: Verify Denominator is Non-Zero

Since direct substitution suggests a non-zero denominator with \( 35 - 10 = 25 \), check that the denominator is indeed non-zero. The expression is not undefined because the denominator cannot be zero at \( x = 5 \).

Step 3: Simplify and Re-Evaluate the Limit

Since the initial computation from direct substitution showed non-zero results, there is no need to simplify further as the division is valid. The function simplifies to \( \frac{50}{25} = 2 \), and hence:\[\lim _{x \rightarrow 5} \frac{3x^2 - 5x}{7x - 10} = 2\]

Step 4: Conclusion and Result

Since the substitution and simplification confirm the limit. The solution result remains valid as \( 2 \).

Key Concepts

Direct Substitution

Direct substitution is the simplest method to determine limits in calculus. It involves directly replacing the variable in the function with the value it approaches. In this exercise, we directly replaced the variable \(x\) with \(5\) in the rational function \(\frac{3x^2 - 5x}{7x - 10}\). When we perform the calculation:

• Numerator: \(3(5)^2 - 5(5)\) equals \(75 - 25 = 50\).
• Denominator: \(7(5) - 10\) equals \(35 - 10 = 25\).

By evaluating these expressions, we see that the function does not lead to an indeterminate form like \(\frac{0}{0}\). Therefore, direct substitution gives a valid result of \(\frac{50}{25} = 2\). This method is straightforward and works well when the function is continuous and not undefined at the given point.

Rational Functions

Rational functions are expressions involving polynomials in both the numerator and the denominator. In mathematical terms, they are of the form \(\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomial expressions. These can sometimes exhibit indeterminate behavior if the denominator equals zero at the point of interest.However, not all evaluations of rational functions at specific limit points lead to division by zero. For instance, in the limit \(\lim _{x \rightarrow 5} \frac{3x^2 - 5x}{7x - 10}\), the denominator at \(x = 5\) is non-zero, simplifying our process considerably.Understanding rational functions allows us to predict their behavior:

• They may simplify to reveal simpler expressions after polynomial division or factorization.
• Non-zero denominators at limit points prevent discontinuity or undefined expressions.

Knowing these properties aids us in rational function limit evaluations without complexity.

Limit Evaluation Steps

Evaluating limits systematically involves key steps that ensure understanding and correctness. Here's a quick guide to the limit evaluation process:Start by evaluating the function via direct substitution:

• Substitute the approaching value into the function for straightforward results.

If direct substitution results in an indeterminacy such as \(\frac{0}{0}\), further techniques such as factoring, expanding, or simplifying may be necessary to resolve it.Verify that the answer makes sense:

• Check that any simplification or algebraic manipulation does not change the function's nature at the point.
• Ensure the denominator is non-zero to avoid division by zero issues.

In our exercise, direct substitution straightaway yielded a valid limit value of 2 without further steps, as the denominator was already non-zero.Ultimately, having a clear procedure for each problem type reduces error and streamlines the evaluation process.