Problem 34 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated limit. \(\lim _{x \rightarrow 1} \frac{x^{3}+1}{2 x^{3}+2}\)

Step-by-Step Solution

Answer

Since the function is continuous at x = 1, we can find the limit by plugging in the value of x, giving us \(\lim_{x \rightarrow 1} \frac{x^3+1}{2x^3+2} = \frac{1}{2}\).

Step 1: Identify indeterminate forms

Let's plug in x = 1 into the function and see if the result is an indeterminate form (0/0 or ∞/∞). \( \frac{(1)^3+1}{2(1)^3+2} = \frac{2}{4} \) The function is not indeterminate; therefore, the limit exists and equals the value of the function at x = 1.

Step 2: Evaluate the limit

Since the function is continuous at x = 1, we can find the limit by plugging in the value of x: \( \lim_{x \rightarrow 1} \frac{x^3+1}{2x^3+2} = \frac{(1)^3+1}{2(1)^3+2} \) Now, evaluate the expression: \( \lim_{x \rightarrow 1} \frac{x^3+1}{2x^3+2} = \frac{1+1}{2+2} = \frac{2}{4} \) So the limit is equal to 1/2. Our final answer is: \( \lim_{x\rightarrow 1} \frac{x^3+1}{2x^3+2} = \frac{1}{2} \)

Key Concepts

Indeterminate Forms

When dealing with limits in calculus, it's crucial to recognize indeterminate forms because they often require special techniques to evaluate. Indeterminate forms are expressions where substitution directly into the limit leaves us with an undefined expression. The most common indeterminate forms are \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). These forms signal that direct substitution of the limit point makes the expression undefined, requiring analysis or manipulation for a clearer form.

In the context of the exercise, when substituting \(x = 1\) into \(\frac{x^3+1}{2x^3+2}\), we find \(\frac{2}{4}\). This result is not an indeterminate form, as the function is well-defined and finite at this point. Since it's neither \(\frac{0}{0}\) nor \(\frac{\infty}{\infty}\), we can comfortably say there's no further need for complicated calculations. This allows us to directly evaluate the function's value at \(x = 1\).

Limit Evaluation

Limit evaluation is a fundamental concept in calculus, where we determine the behavior of a function as it approaches a particular point. To evaluate limits, substitution is often the first step. If the function is continuous at that point, and it does not result in an indeterminate form, you can simply substitute the value directly.

In the exercise given, by plugging in the value \(x = 1\) directly into the function \(\frac{x^3+1}{2x^3+2}\), substitution gives a clear and definite result \(\frac{1}{2}\). Because a straightforward substitution yields a non-indeterminate and finite outcome, it confirms that this is indeed the limit as \(x\) approaches 1. This method is quick and efficient, saving time when dealing with complex expressions only when necessary.

Continuity in Functions

Understanding continuity helps us approach limits with more confidence. A function is continuous at a point if there is no interruption or gap in its graph at that point. Formally, a function is continuous at \(x = a\) if the limit as \(x\) approaches \(a\) is equal to the function's value at \(a\).

This characteristic of continuity is beneficial, particularly in the exercise above. Since \(\frac{x^3+1}{2x^3+2}\) is a rational function and rational functions are continuous wherever they are defined, we can directly calculate the limit at \(x = 1\). This explains why substituting \(x = 1\) without further tricks or transformations accurately gives the function's limit. Recognizing continuity removes the necessity of applying limit theorems or algebraic manipulations for simplification purposes.