Question

Determine the following limits and justify your answers. $$\lim _{x \rightarrow-1}\left(x^{2}-4+\sqrt[3]{x^{2}-9}\right)$$

Short answer

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

Step-by-step solution

Identify the given expression and the limit

We are asked to find the following limit: $$\lim_{x \rightarrow -1} \left(x^2-4+\sqrt[3]{x^2-9}\right)$$

Check if we can directly substitute the value into the expression

We will first attempt to obtain a solution by direct substitution of the value -1 in place of x: $$(-1)^2 - 4 + \sqrt[3]{(-1)^2 - 9} = 1 - 4 + \sqrt[3]{1-9}$$ Notice that there are no undefined expressions in this case (for example, division by zero or square root of a negative number), so we can proceed to compute the expression.

Compute the expression

Now, we need to compute the given expression with x = -1: $$ 1 - 4 + \sqrt[3]{1-9} = 1 - 4 + \sqrt[3]{-8} $$ Since the cube root of -8 is -2, the expression becomes: $$1 - 4 + (-2) = -5$$

Determine the limit by substitution

We found out that the expression has a well-defined value at x = -1, which means that the limit can be obtained by direct substitution. Therefore, the limit of the given expression as x approaches -1 is: $$\lim_{x \rightarrow -1} \left(x^2-4+\sqrt[3]{x^2-9}\right) = -5$$

Key concepts

Direct Substitution

A very convenient and straightforward method to find limits in calculus is through direct substitution. This technique involves simply replacing the variable in the expression with the value it approaches. If the resulting expression is defined and finite, the limit is easily determined.
Direct substitution is most effective when the function is continuous at the point of evaluation. For instance, in the example provided, substituting \( x = -1 \), results in a defined expression as there is no division by zero or taking an even root of a negative number. Thus, the function:

• Is continuous at the point where you substitute.
• Results in a valid numerical expression.

Always ensure that the substitution does not lead to undefined operations, allowing for the feasible use of this method.

Expression Evaluation

After determining that direct substitution is possible, the next step is to evaluate the expression. For this specific problem, the expression is
\[ x^2 - 4 + \sqrt[3]{x^2 - 9} \]
To evaluate, substitute \( -1 \) into the expression:

• Calculate \( (-1)^2 = 1 \)
• Subtract 4 from 1, giving \( 1 - 4 = -3 \)
• Simplify the cube root component \( \sqrt[3]{(-1)^2 - 9} \) to \( \sqrt[3]{-8} \).

Keep track of negative signs and operations as you simplify the expression, ensuring each arithmetic operation is accurately performed, leading to a final evaluation of the expression without errors.

Cube Roots

Understanding cube roots is vital in evaluating expressions like:
\( \sqrt[3]{-8} \). The cube root of a number \( a \) is that number \( b \) which when multiplied by itself twice results in \( a \): \( b^3 = a \).
Unlike square roots, cube roots of negative numbers result in negative values. Thus,

• The cube root of -8 is -2 because \((-2)^3 = -8\).
• This concept allows each cube root to be directly simplified, even for negative numbers.

Ensure familiarity with how cube roots operate, especially in context with negative results, which are common in calculus problems involving limits and derivatives.