Problem 14 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

The limit is 6.

Step 1: Substitute the Limit Value

First, attempt to substitute the value that \( x \) approaches directly into the function. Here, we substitute \( x = 7 \) into the expression to see if it yields a value.\[\frac{7^2 - 7}{2(7) - 7} = \frac{49 - 7}{14 - 7} = \frac{42}{7} = 6\]Since \( \lim _{x \rightarrow 7} \frac{x^{2}-x}{2 x-7} \) directly gives you a real number, no further simplification is required.

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Key Concepts

Direct Substitution Method

The direct substitution method is one of the simplest ways to find limits. It's like testing the waters by plugging in the value that "x" is approaching directly into the function. Doing so can sometimes resolve the limit question immediately. In our exercise, we straightforwardly substituted \( x = 7 \) into \( f(x) = \frac{x^{2}-x}{2 x-7} \).This substitution unfolded rather smoothly because it did not lead to any complications, like division by zero or indeterminate forms. The calculation went as follows:

• Substitute: \( x = 7 \) in \( f(x) \).
• Calculate: \( \frac{49-7}{14-7} \).
• Result: \( \frac{42}{7} = 6 \).

Since we got a clear number, 6, without any hurdles, the direct substitution method served its purpose effectively. However, direct substitution may not always work so seamlessly if the denominator equals zero, at which point we might need other techniques.

Finite Limit Evaluation

Finite limit evaluation means that when calculating the limit of a function as \( x \) approaches a certain value, the result is a finite, well-defined number. In our example, we aimed to find the limit of the rational function \( \frac{x^{2}-x}{2x-7} \) as \( x \) approaches 7.Using the direct substitution, we achieved a finite result of 6. This finite outcome tells us that the behavior of the function near the point \( x = 7 \) is stable and predictable. Finite limits are quantitative. This means they provide concrete values which gives us firm conclusions about function behavior around specific points without resulting in infinity or undefined results.If the function yielded an infinite value, such as \(+\infty\) or \(-\infty\), or if it was an indeterminate form such as \( \frac{0}{0} \), we would need further analysis. But in our case, the function settled at a tidy number — 6.

Rational Function Limits

Rational functions are fractions where both the numerator and the denominator are polynomials. The limit of a rational function as \( x \) approaches a specific value can often be found using direct substitution, provided it doesn't lead to a zero denominator.In our exercise, the given function, \( \frac{x^{2}-x}{2x-7} \), is a classic rational function. We successfully applied the direct substitution method without much hassle to find its limit value as \( x \) approaches 7.When working with rational function limits, it’s helpful to remember:

• If direct substitution does not result in division by zero, it's the fastest method.
• If substitution results in \( \frac{0}{0} \), consider simplifying the expression, possibly factoring where applicable.
• If it leads to infinity, you might need to interpret the behavior as \( x \) approaches the value.

In our situation, substituting the number 7 directly gave us a non-zero and finite result, showcasing that not all rational function limits require elaborate manipulation.