Problem 50 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The limit of the function $(x^2 - 4 + \sqrt[3]{x^2 - 9})$ as $x$ approaches $-1$ is $-5$.

Step 1: Substitute the value of x

Since the function is continuous at x = -1, we can directly substitute the value of x into the given function: $$\lim _{x \rightarrow-1}\left(x^{2}-4+\sqrt[3]{x^{2}-9}\right) =((-1)^{2}-4+\sqrt[3]{(-1)^{2}-9})$$

Step 2: Simplify the expression

Now, we simplify the expression: $$((-1)^{2}-4+\sqrt[3]{(-1)^{2}-9}) =(1-4+\sqrt[3]{1-9})$$

Step 3: Calculate the limit

Since the expression is now simplified, we can compute the limit: $$\lim _{x \rightarrow-1}\left(x^{2}-4+\sqrt[3]{x^{2}-9}\right) =(1-4+\sqrt[3]{1-9})=(-3+\sqrt[3]{-8})$$

Step 4: Evaluate the cubic root

Finally, we evaluate the cubic root of a negative number: $$-3+\sqrt[3]{-8}=-3+(-2)$$

Step 5: Get the final answer

The final step is to combine the remaining terms: $$-3+(-2)=-5$$ So the limit of the given function is: $$\lim _{x \rightarrow-1}\left(x^{2}-4+\sqrt[3]{x^{2}-9}\right)=-5$$

Key Concepts

Continuity

When faced with evaluating a limit, continuity plays a crucial role. In simple terms, a function is continuous at a point if there is no interruption in its graph at that point.
This means you can draw it without lifting your pencil. If a function is continuous at a certain value of \(x\), you can directly substitute \(x\) into the function to find the limit.

• Benefits include straightforward calculations and less need for complex algebraic manipulations.
• Being able to plug directly into a function at the limit point simplifies finding limits.

In our problem, the expression \(x^2 - 4 + \sqrt[3]{x^2 - 9}\) is continuous at \(x = -1\). This is because the polynomial \(x^2\) is continuous everywhere and the cubic root is continuous as there is no division by zero or other discontinuity at \(x = -1\).
This allows us to substitute \(x = -1\) directly into the function, making the evaluation of limits much more seamless.

Cubic roots

Cubic roots may seem tricky, but they are easier once you break them down. The cubic root of a number \(y\), denoted \(\sqrt[3]{y}\), is a number that, when multiplied by itself twice, gives \(y\).
Cubic roots can be taken of both positive and negative numbers:

• For positive numbers, the cubic root is also positive.
• For negative numbers, the cubic root is negative, since the product of three negatives is negative.

In this problem, the value inside the cubic root is \(-8\) because substituting \(-1\) into \(x^2 - 9\) gives \(-8\). The cubic root \(\sqrt[3]{-8}\) equals \(-2\) because \((-2) \times (-2) \times (-2)\) equals \(-8\). Understanding how to simplify and calculate cubic roots like this helps in solving complex-looking problems by simplifying them into more manageable parts.

Algebraic simplification

Algebraic simplification is the process of making complicated expressions more manageable and easier to understand or compute. By doing this, you can easily evaluate limits, derivatives, or other mathematical functions.Here's how you can approach simplification:

• First, substitute any known values into the expression, such as the value of \(x\) in a limit.
• Next, perform arithmetic to simplify the expression as much as possible.
• This includes finding and simplifying roots like cubic roots when necessary.

In this problem, after substituting \(x = -1\) into \(x^2 - 4 + \sqrt[3]{x^2 - 9}\), we got \(1 - 4 + \sqrt[3]{1 - 9}\), which simplifies to \(-3 + \sqrt[3]{-8}\).
Then, knowing \(\sqrt[3]{-8}\) simplifies further to \(-2\), the final computation \(-3 + (-2) = -5\) is achieved. Breaking down these steps simplifies the process, making it easier to reach an accurate answer quickly.