Problem 14 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 2} \sqrt[3]{12 x+3} $$

Step-by-Step Solution

Answer

The limit of the function \( \lim _{x \rightarrow 2} \sqrt[3]{12 x+3}\) is 3

Step 1: Understand the function

The function given is \( \lim _{x \rightarrow 2} \sqrt[3]{12x+3} \). This function describes a limit which means it is instructing us to find the value of the function as \(x\) approaches 2.

Step 2: Substitution of Value

We substitute \(x=2\) into the function. This would result in \( \sqrt[3]{12(2) + 3} = \sqrt[3]{27} \).

Step 3: Calculating the Cube Root

Calculate the cube root of 27 which gives 3.

Key Concepts

Limits and Continuity

Understanding the concept of limits is crucial in calculus because it deals with values that functions approach as the input gets closer to a certain point. The concept of limits is integral to the study of continuity. A function is continuous at a point if there are no breaks, jumps, or holes at that point. In other words, you should be able to draw it without lifting your pencil. When we say that the limit of a function equals a certain value as the input approaches a particular point, we're asserting that the function's value gets indefinitely close to the limit value.

For limits to exist and for a function to be continuous at a specific point, three conditions must be met: the function must be defined at that point, the limit must exist as the input approaches that point, and indeed, the function's value at that point must equal the limit value. The provided exercise demonstrates a continuous function where the limit as x approaches 2 can be determined and happens to be a real number. This indicates that there's no break at x=2 and thus the function is continuous at this point.

Evaluating Continuity

To assess continuity at x=2 for the function \( \sqrt[3]{12x+3} \) you can check whether the function is defined, if the limit exists, and if the function's value matches the limit at x=2. All three criteria are met which confirms the function is continuous at x=2.

Evaluating Limits

Evaluating limits often involves figuring out what happens to a function as the input value approaches, but does not necessarily reach, a certain point. In many cases, you can simply substitute the value of x into the function to find the limit. This direct substitution method is often the first approach one should take. If direct substitution results in a value without a mathematical error, like division by zero, then the limit can be evaluated through substitution.

In the given exercise, we practice evaluating the limit of a cube root function by substituting x with 2. Doing so gives us \( \sqrt[3]{27} \), which clearly equals 3. However, if substitution led to an undefined expression or an indeterminate form like 0/0, further limit solving techniques such as factoring, rationalizing, or applying L'Hôpital's rule might be necessary.

Direct Substitution

Consider the direct substitution method as an initial approach to evaluate limits. If \( f(x) \) simplifies to an exact value when \( x \) is replaced with the given point, then \( \lim_{{x \to a}} f(x) = f(a) \). As done in this example, substituting x=2 into \( \sqrt[3]{12x+3} \) we found that the limit is simply 3.

Cube Roots

Cube roots are the inverses of cubic functions, just like square roots are the inverses of quadratic functions. The cube root of a number x, denoted as \( \sqrt[3]{x} \), is a value that, when multiplied by itself three times, gives the original number. Cube roots can handle both positive and negative values, since cubing either a positive or negative number results in a positive or negative answer respectively.

The exercise at hand requires calculating the cube root of 27 which is straight forward since it’s a perfect cube, meaning the answer is an integer. The cube root of 27 is 3 because \( 3 \times 3 \times 3 = 27 \). In contrast to square roots, which only take non-negative inputs to return real numbers, cube roots allow negative inputs as well, providing a broader range of solutions.

Working with Cube Roots

Always remember that unlike square roots, cube roots do not restrict the domain to non-negative numbers. For instance, \( \sqrt[3]{-8} = -2 \) because \( (-2) \times (-2) \times (-2) = -8 \), highlighting how cube roots accommodate all real numbers as inputs.