Problem 25 - Calculus and its applications

Question

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 2} \frac{x^{3}-8}{2-x} $$

Step-by-Step Solution

Answer

The limit is \(-12\).

Step 1: Direct Substitution

To start, substitute \(x = 2\) into the function to check if it results in an indeterminate form. Calculate \((2)^3 - 8\) which equals \(8 - 8 = 0\). Substitute \(2\) in the denominator: \(2 - 2 = 0\). Thus, the substitution leads to \(\frac{0}{0}\), confirming it's an indeterminate form.

Step 2: Rewrite the Function

Notice the numerator is a difference of cubes: \(x^3 - 8 = (x - 2)(x^2 + 2x + 4)\). Therefore, rewrite the limit expression as: \[ \lim_{x \to 2} \frac{(x-2)(x^2 + 2x + 4)}{2-x} \] The denominator \(2 - x\) can be rewritten as \(-(x-2)\).

Step 3: Simplify the Expression

After rewriting the denominator, the expression becomes: \[ \lim_{x \to 2} \frac{(x-2)(x^2 + 2x + 4)}{-(x-2)} \] Cancel \((x-2)\) from the numerator and denominator, knowing that \(xeq 2\): \[ \lim_{x \to 2} - (x^2 + 2x + 4) \]

Step 4: Evaluate the Simplified Limit

Now substitute \(x = 2\) into the simplified expression \(-(x^2 + 2x + 4)\): \(-(2)^2 -2(2) -4 = -(4 + 4 + 4) = -12\). After simplification, the limit is evaluated as \(-12\).

Key Concepts

Indeterminate Forms

When dealing with limits in calculus, sometimes you encounter what is known as an "indeterminate form". This happens when substituting a value into a function results in forms like \(\frac{0}{0}\), which don't clearly define a limit or even indicate the behavior of the function near that point.

• Common indeterminate forms include \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\).
• These forms require further analysis to determine a meaningful limit or determine if a limit exists at all.

In our original problem, after substituting \(x = 2\), it resulted in \(\frac{0}{0}\). This signals the need for additional steps to evaluate the limit correctly. Recognizing these forms is key since they guide you towards methods like simplification or transformation to resolve them.

Simplifying Expressions

Simplifying an expression involves rewriting it in a form that is easier to work with, particularly for evaluating limits. In our problem, we began by identifying an algebraic component that could be factored, namely the numerator.

• The numerator \(x^3 - 8\) is recognized as a difference of cubes.
• By factoring it into \((x - 2)(x^2 + 2x + 4)\), we have exposed a common factor that also appears in the denominator.

This step is crucial, as it allowed us to cancel out terms, simplifying our original complex expression to a basic polynomial, which is much easier to evaluate for limits. Simplifying helps in not only solving the given limit but also understanding the overall behavior of the function at a particular x-value by reducing computational complexity.

Difference of Cubes

Identifying special algebraic forms like the difference of cubes is integral in simplifying expressions. A difference of cubes formula is expressed as: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\].

• In the exercise, \(x^3 - 8\) can be rewritten using this formula by recognizing it as \((x^3 - 2^3)\), allowing us to factor it.
• This gives \((x - 2)(x^2 + 2x + 4)\), matching the factor \(x - 2\) in the denominator once rewritten as \(-(x - 2)\).

Factoring using the difference of cubes can simplify the often-complicated higher degree polynomials into products of simpler expressions, making it easier to see where terms cancel out and simplify further.

Evaluating Limits

After applying algebraic simplifications, the next step is to evaluate the limit of the simplified expression. Post-simplification, the function may no longer exhibit an indeterminate form at the point of interest.

• With our problem, simplifying the expression led us to evaluate \[-(x^2 + 2x + 4)\].
• Substituting \(x = 2\) results in \(-(4 + 4 + 4) = -12\).

The simplicity of polynomial evaluation means we can straightforwardly substitute and compute without the earlier complexities of an indeterminate form. Evaluating limits is the ultimate goal, providing insight into the function's behavior near specific values of \(x\). In this case, we confirmed that the limit was \(-12\) as \(x\) approaches \(2\).