Problem 13 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow - 2 } ( x - 6 ) ^ { 2 / 3 }$$

Step-by-Step Solution

Answer

Therefore, the solution to \(\lim _ { x \rightarrow - 2 } (x - 6) ^ {2 / 3}\) is \(4\).

Step 1: Substitute x in the function

According to the limit's substitution principle, first, we need to substitute \(x = -2\) into the function \((x - 6)^{2/3}\). So our function will change into \((-2 - 6)^{2/3}\).

Step 2: Solving the Expression

After substituting \(x=-2\) into the function, let's solve the equation \((-2 - 6)^{2/3}\). First, calculate the result of \(-2 - 6\), which equals \(-8\). Now the equation becomes \((-8)^{2/3}\.

Step 3: Calculating the Cube Root

Next, let's simplify the equation further. The cube root of -8 is -2. So, \((-8)^{2/3}\) equals to \((-2)^2\), which is \(4\). Therefore, the limit of the function \((x - 6)^{2/3}\) as \(x\) approaches \(-2\) is \(4\).

Step 4: Graphical Representation

When graphing the function \((x - 6)^{2/3}\), we will observe that as \(x\) approaches \(-2\), the function approaches \(4\). This graphical representation confirms the limit found algebraically.

Key Concepts

Limit by Substitution

Understanding the concept of limit by substitution is a foundational skill in calculus. It involves directly substituting the value into the function when finding the limit. This technique works effectively for functions that are continuous at the given point. In our exercise, this means replacing

• \(x = -2\)
• into the expression \((x-6)^{2/3}\).

By doing so, we change the expression to

• \((-2-6)^{2/3}\).

This approach is straightforward and a first step in evaluating limits without further algebraic manipulation. Substitution helps in checking if a function behaves normally at a point or if it requires more advanced methods.

Algebraic Manipulation

After substitution, algebraic manipulation helps simplify the expression to find the exact limit. In the exercise,

• we initially solve for \(-2 - 6\), which simplifies to \(-8\).

This results in the expression

• \((-8)^{2/3}\).

Algebraic manipulation involves calculating powers and roots in this case. By finding the cube root of \(-8\), which is \(-2\), and then squaring the result, we simplify

• \((-8)^{2/3}\) to \((-2)^2 = 4\).

These steps ensure we approach the limit methodically, confirming the behavior of the function at the specific point. Understanding these manipulations is critical for tackling more complex limits.

Graphical Representation

Graphical representation offers a visual perspective on limits. Plotting the function provides an understanding of how it behaves around certain points. For the function

• \((x-6)^{2/3}\),
• we analyze its curve as \(x\) approaches \(-2\).

The graph should show the function approaching the value of \(4\). Visual confirmation through graphs supports the algebraic calculations and enriches comprehension. This method highlights whether the limit is continuous or if any discontinuity exists. Graphs are beneficial in identifying trends, asymptotic behaviors, and verifying calculated limits.

Cube Roots

Cube roots play a crucial role in simplifying expressions such as \((-8)^{2/3}\). A cube root is a value that, when multiplied by itself twice, yields the original number. In our case:

• The cube root of \(-8\) is \(-2\).

Calculating the cube root first and then applying the power provides the simplest form of expressing the result of an exponent like \(2/3\).

• This simplifies the expression to \((-2)^2 = 4\).

Understanding cube roots is vital, as they frequently appear in calculus when dealing with radical expressions. Mastering this concept allows for accurate simplification and solution of complex limit problems.