Problem 48 - Calculus Early Transcendentals

Question

Evaluate each limit and justify your answer. $$\lim _{x \rightarrow-1}\left(x^{2}-4+\sqrt[3]{x^{2}-9}\right)$$

Step-by-Step Solution

Answer

Answer: The limit of the function as x approaches -1 is -5.

Step 1: Identify the indeterminate form, if any

As x approaches -1, let's analyze the three parts of the function: 1. Polynomial: x^2 - 4 2. Rational: cube root of (x^2 - 9) Both the polynomial and rational parts can be evaluated directly by substituting x = -1 into the expression.

Step 2: Evaluate the polynomial part of the function

We will evaluate the polynomial part (x^2 - 4) by substituting x = -1: \((-1)^2 - 4 = 1 - 4 = -3\)

Step 3: Evaluate the rational part of the function

Now we will evaluate the cube root of (x^2 - 9) by substituting x = -1: \(\sqrt[3]{(-1)^2 - 9} = \sqrt[3]{1 - 9} = \sqrt[3]{-8} = -2\)

Step 4: Combine the evaluated parts to find the limit

Now, combining the evaluated polynomial and rational parts to get the overall limit: $$\lim _{x \rightarrow-1}\left(x^{2}-4+\sqrt[3]{x^{2}-9}\right) = -3 + (-2) = -5$$ So, the limit of the given function as x approaches -1 is -5.

Key Concepts

Indeterminate Forms

When evaluating limits in calculus, we often encounter expressions that do not readily give a clear answer and are known as indeterminate forms. These are mathematical expressions that cannot be exactly determined from the information given. In the context of the given exercise, the indeterminate form does not appear explicitly; however, in more complex functions, they can manifest as 0/0, ∞/∞, 0×∞, ∞ - ∞, 1^∞, 0^0, and ∞^0.

Indeterminate forms require special treatment because simply plugging in the values won't provide the correct limit. Techniques like factoring, conjugation, rationalization, or applying L'Hôpital's Rule are used to resolve these forms and accurately evaluate the limit.

Understanding indeterminate forms is crucial in calculus because it helps identify when a limit might need a closer look and further manipulation before it can be evaluated effectively.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of this is the expression presented in the exercise, \(x^2 - 4\), which is a simple polynomial of degree 2, also called a quadratic function.

When evaluating limits involving polynomial functions, as x approaches a particular value, you can typically substitute that value directly into the polynomial to find the limit. This is because polynomial functions are continuous over their entire domain, which means there are no breaks, holes, or jumps in their graph. In our exercise, the polynomial part evaluates neatly to -3 when \(x = -1\). This direct substitution property makes polynomial functions straightforward to handle in limit problems.

Rational Expressions

Rational expressions are ratios of two polynomial functions. Unlike polynomial functions, they can have discontinuities like holes or vertical asymptotes, potentially complicating limit evaluation. However, in the exercise, we encounter a slightly different concept—a cube root function, which is a type of radical expression and behaves similarly to rational expressions for calculus purposes.

Radical expressions, particularly roots other than the square root, can be interpreted similarly to rational expressions when considering limits. For instance, the cube root part of the function in the exercise, \(\sqrt[3]{x^2 - 9}\), approaches a specific value as x approaches -1, in this case -2. This is because radical functions are also continuous wherever they are defined, allowing for direct substitution as was done with the polynomial. It's important for students to recognize that rational and radical expressions can be treated with similar methods when evaluating limits, particularly when expressions do not create an indeterminate form.