Problem 30 - Calculus: Early Transcendentals

Question

Compute the following limits. $$ \lim _{x \rightarrow 2} \frac{x^{3}-6 x-2}{x^{3}+4} $$

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{2}\).

Step 1: Substitute the Limit Value

Start by substituting the limit value, \( x = 2 \), directly into the function. This gives:\[\lim _{x \rightarrow 2} \frac{x^{3}-6x-2}{x^{3}+4} = \frac{2^{3}-6 \cdot 2-2}{2^{3}+4}\]

Step 2: Calculate the Numerator and the Denominator

Calculate both the numerator and the denominator with \( x = 2 \):For the numerator: \[2^3 - 6 \cdot 2 - 2 = 8 - 12 - 2 = -6\]For the denominator:\[2^3 + 4 = 8 + 4 = 12\]

Step 3: Evaluate the Fraction

Substitute the calculated values of the numerator and the denominator into the fraction:\[\frac{-6}{12} = -\frac{1}{2}\]

Step 4: Confirm No Further Simplification or Indeterminate Forms

As both the numerator and the denominator are simple constants after substitution with no zeros or indeterminate forms (\(0/0\), \(\infty/fty\)), the calculated limit is valid.

Key Concepts

Limit Evaluation

Limit evaluation is an essential concept in calculus. It allows us to understand the behavior of functions as they approach specific points or infinity. When evaluating limits, we're interested in what happens to the value of the function as the input variable gets very close to a given number. Limits help decipher the behavior of function values even when they aren't explicitly defined at certain points. This concept is foundational for understanding continuity, derivatives, and integrals.

In the provided problem, we evaluate the limit of a function as the variable approaches 2. **Evaluating limits** can involve different strategies like direct substitution or more advanced techniques if the function doesn't conveniently resolve to a number. Methods such as substituting the limit point directly and then solving are quite common for simple cases. More complex scenarios might involve algebraic manipulation or the application of special limit rules.

Substitution Method

The substitution method for evaluating limits is one of the simplest ways to find the value of a function as it approaches a certain point. With this method, we substitute the value into the function directly. If the function is well-defined at that point, this yields the limit.

In the original exercise, by substituting the limit value of 2 into the function, we were able to find the limit. When we plug in 2 into both the numerator and the denominator, we calculate:

• The numerator becomes: \(2^3 - 6 \cdot 2 - 2 = -6\)
• The denominator becomes: \(2^3 + 4 = 12\)

After substitution, the function simplifies to a straightforward fraction: \(\frac{-6}{12} = -\frac{1}{2}\). This method, however, only works cleanly if the substitution doesn't lead to undefined expressions like dividing by zero or encountering indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when substitution into a limit yields an ambiguous result such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These forms do not provide clear information about the behavior of the function and indicate a need for further analysis.

In this exercise, after substitution, the result was straightforward since we obtained a defined fraction \(\frac{-6}{12}\). However, if the substitution resulted in an indeterminate form, we might need additional techniques like factoring, rationalization, or L'Hôpital's Rule to resolve the limit. It's crucial to recognize and distinguish these forms, as they indicate that the direct approach of substitution will not work without further manipulation to simplify the expression first. Understanding indeterminate forms is vital for tackling more complex limits effectively.