Problem 15 - Thomas Calculus

Question

Find the limits in Exercises \(11-22\) $$\lim _{x \rightarrow 2} \frac{2 x+5}{11-x^{3}}$$

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Substitute the Limit Value

First, substitute the limit value \(x = 2\) directly into the function to see if it is a straightforward evaluation or if adjustments are needed:\[ f(x) = \frac{2x + 5}{11 - x^3} \]Substituting \(x = 2\):\[ \frac{2(2) + 5}{11 - (2)^3} = \frac{4 + 5}{11 - 8} = \frac{9}{3} \]

Step 2: Simplify the Expression

Simplify the expression from the substitution:\[ \frac{9}{3} = 3 \]

Step 3: Verify the Limit Existence

Check if direct substitution leads to an indeterminate form or if division by zero occurs. In this problem, substitution directly provides a finite result without division by zero or need for further simplification. Since the substitution gives a valid number, there is no need for other limit finding techniques such as factoring, rationalization, or L'Hôpital's Rule.

Key Concepts

Direct Substitution Method

When dealing with calculus limits, the "Direct Substitution Method" is a straightforward approach. It involves taking the value that the variable approaches and directly substituting it into the function. This is the first step to see if we can easily find the limit. It is often the quickest way to determine the limit, but it does not always work.

Here's what you do:

• Take the limit value, which is what "x" approaches in the limit notation, and plug it right into the function.
• If you substitute and get a real number, congrats! Your job is done because the limit is reached.
• If you get a bizarre result like division by zero or math involving undefined terms, don't worry. There might be another method needed. In such cases, the direct substitution does not work, and you need to try other techniques.

For the given exercise, using direct substitution provided a clear result: substituting "2" into the function directly solved it, giving us the limit of 3. There was no division by zero or undefined expressions encountered, making it a perfect scenario for this method.

Limit Evaluation

Limit evaluation takes calculus into a realm where we explore the behavior of functions as variables approach specific points, even when it's a bit tricky to compute directly. It involves a couple of steps and thoughts to ensure you're approaching the process correctly.

Start by replacing the variable with the approaching value:

• Substitute the target value directly into the function first, which is what we learned as the direct substitution.
• If the substitution gives you a clean mathematical expression, as it did in our example when we found \(\frac{9}{3}\), it means the limit is straightforward and approachable.
• Should direct substitution not work, maybe due to a zero-denominator or more complex scenario, you might need higher-level techniques like factoring, rationalizing, or leveraging L'Hôpital's Rule.

By evaluating limits, we can understand stability and predictability of functions around points that might not be easily accessible by just substitute and simplify.

Indeterminate Forms

Indeterminate forms appear in calculus when initial substitution does not give a straightforward result. These forms indicate situations that are seemingly undefined and require further analysis.

Common scenarios for indeterminate forms include expressions that result in:

• \(\frac{0}{0}\)
• \(\frac{\infty}{\infty}\)
• \(0 \times \infty\)
• \(\infty - \infty\)
• \(0^0, 1^\infty, \) and \(\infty^0\)

In these cases, the result isn't clear, so mathematical techniques are needed. Methods such as factoring, L'Hôpital’s Rule, or even trigonometric identities may be applied to simplify the expression and resolve the indeterminate form.

In our initial exercise, direct substitution gave us a definite number, avoiding the issue entirely. However, if the substitution had resulted in something like \(\frac{0}{0}\), we would have to explore these advanced methods to solve it correctly. Recognizing indeterminate forms is key as they signal the need for deeper analysis and other calculus tools.