Question

Finding a Limit In Exercises \(41-46,\) write a simpler function that agrees with the given function at all but one point. Then find the limit of the function. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow 2} \frac{x^{3}-8}{x-2} $$

Short answer

The limit as \(x\) approaches 2 of the function given by \(\frac{x^{3}-8}{x-2}\) is 12.

Step-by-step solution

Simplify the Function

Begin by simplifying the function. Here, you will recognize the numerator \(x^{3} - 8\) as a difference of cubes. The algebraic identity for a difference of two cubes \(a^{3} - b^{3}\) is \((a - b)(a^{2} + ab + b^{2})\). Apply this identity to rewrite the numerator \((x - 2)(x^{2} + 2x + 4)\). Then the function can be written as \(\frac{(x - 2)(x^{2} + 2x + 4)}{x - 2}\). Notice that the term \((x - 2)\) can be cancelled out from the numerator and the denominator.

Cancel Common Terms and Simplify

After cancelling the term \((x - 2)\), the function becomes \(f(x) = x^{2} + 2x + 4\).

Find the Limit

The limit of the function as \(x\) approaches 2 can now be found by substituting \(x = 2\) into the simplified function. Doing so gives \( f(2) = (2)^{2} + 2*(2) + 4 = 4 + 4 + 4 = 12\). Thus, the limit of the function as \(x\) approaches 2 is 12.

Confirm the Result

To confirm this result, you can use a graphing utility to graph the function \(x^{2} + 2x + 4\) and inspect the behavior of the function as \(x\) approaches 2. The graph should approach the y-value of 12 as \(x\) approaches 2, confirming that the limit is indeed 12.

Key concepts

Difference of Cubes

The difference of cubes refers to a special algebraic factorization formula used for simplifying expressions like \(x^3 - a^3\), which is the difference between two cube terms. In calculus, this technique is often employed when dealing with limits where polynomial expressions with cubic terms are involved.

For any two variables \(a\) and \(b\), the formula for factoring the difference of cubes is: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]. This allows students to break down complex cubic expressions into simpler, easier-to-manage factors. For example, in the given exercise, \(x^3 - 8\) can be viewed as a difference of cubes where \(a = x\) and \(b = 2\), since \(2^3 = 8\). By applying the formula, the expression is simplified, which is crucial for finding limits when the direct substitution of the value into the original function is not possible.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as its input approaches a particular point, but does not necessarily reach that point. The limit question asks: What value does the function approach as the variable gets arbitrarily close to a certain number?

For example, to find \(\lim_{x \rightarrow a} f(x)\), one typically simplifies the function \(f(x)\) and then evaluates it at \(x = a\), if possible. If direct evaluation leads to an indeterminate form like \(\frac{0}{0}\), algebraic techniques like factoring are used to simplify the expression, as seen with the difference of cubes in the exercise. Understanding how to compute limits is essential because it sets the stage for future topics like continuity, derivatives, and integrals in calculus.

Graphing Utility in Calculus

A graphing utility in calculus is a powerful tool that can visually display the behavior of a function, particularly near points of interest, such as where the limit is being computed.

Graphing the function allows students to confirm the results they obtained algebraically by providing a visual representation. For instance, plotting the function \(f(x) = x^2 + 2x + 4\) near the point \(x = 2\) will show how the function's value gets closer to 12 as \(x\) approaches 2. This helps in understanding concepts like continuity and asymptotic behavior as well, making a graphing utility an indispensable resource in the study of calculus.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a skill that makes evaluating functions, including finding limits, much more manageable. This process involves combining like terms, factoring expressions, expanding products, and canceling common terms.

In the exercise, simplification starts with recognizing the difference of cubes and then canceling the common \(x - 2\) term in the numerator and denominator. The expression reduces to a simpler quadratic expression, which is easier to evaluate as \(x\) approaches a particular value. By mastering simplification techniques, students can solve more complex problems by rendering them down to their basic forms, thus making the overall process of analyzing functions more streamlined and systematic.