Problem 16 - Precalculus with Limits

Question

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{a \to -4} \dfrac{a^3+64}{a+4}$$

Step-by-Step Solution

Answer

The limit of the function as a approaches -4 is 48

Step 1 - Factorize the numerator

To eliminate the term causing the discontinuity, we factorize the cube term in the numerator. This is done using the formula \(a^3+b^3 = (a + b)(a^2 - ab + b^2)\). Thus, \(a^3 + 64 = (a+4)(a^2-4a+16)\). Therefore, the function becomes \(f(a) = \dfrac{(a+4)(a^2 - 4a + 16) }{a+4}\)

Step 2 - Simplify the function

The terms \(a+4\) in the numerator and the denominator will cancel out each other. Hence, the simplified expression for the function is \(f(a) = a^2 - 4a + 16\). We can now substitute -4 into this simplified expression to find the limit.

Step 3 - Evaluate the limit

We now substitute the limit value into the simplified function: \( \lim_{a \to -4} f(a) = (-4)^2 - 4*(-4) + 16 = 16 + 16 + 16 = 48\). So the limit as a approaches -4 is 48

Step 4 - Verify result graphically

To verify this result, you would graph both the original function and the simplified function using a graphing utility. As a approaches -4 from both sides, the y-value should approach 48, which confirms our calculated limit. Note - the graphical verification step isn't shown here as this description requires text-only responses. It should be done on a graphing calculator or similar tool

Key Concepts

Limit Evaluation

Understanding limits is crucial in calculus, as they help us explore the behavior of functions as they approach specific input values. In this exercise, we are evaluating the limit of the function \( \lim_{a \to -4} \dfrac{a^3+64}{a+4} \). This tasks us with finding the behavior of this fraction when \( a \) gets very close to -4.

Finding limits requires careful consideration. Direct substitution of -4 into the expression causes a division by zero, which is undefined. This will often signal a discontinuity that needs to be resolved. Let's look at how to handle such cases with further mathematical techniques.

Factorization

Factorization helps us transform complex expressions into more manageable ones. Here, we need to factor the cube on the numerator. This simplifies the expression significantly.

To break down \( a^3 + 64 \), we use a basic algebraic identity: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). With \( b = 4 \), the expression \( a^3 + 64 \) becomes \((a+4)(a^2 - 4a + 16)\).
This clever step turns our problem into a simpler one, allowing us to identify and remove any problematic factors, which we will explore next.

Discontinuity Elimination

Once we've factorized the expression, we can eliminate terms responsible for discontinuity. In this case, that means removing the shared \( a+4 \) factor from both the numerator and the denominator.

This operation turns \( \dfrac{(a+4)(a^2 - 4a + 16)}{a+4} \) into \( a^2 - 4a + 16 \). This resulting expression no longer has a problem when \( a \) equals -4, as substituting -4 is now valid.

Eliminating discontinuity allows the limit to be calculated straightforwardly. The function is now continuous and defined at our limit point, so we can proceed to evaluate the limit without any struggles.

Graphical Verification

Graphical verification provides a visual confirmation of our limit findings. While algebra offers the technical solution through calculations, a graphing utility makes it tangible.

By graphing both the original and simplified functions, we can observe the behavior as \( a \) approaches -4. Look for the value that the function's output (y-value) is reaching when the input (x-value) nears -4.

• Verify that when approaching -4 from either side, the output approaches our calculated limit of 48.
• This step reassures us that we've correctly simplified the function and removed discontinuities.

A graph essentially becomes the final stamp of correctness, visually cementing our mathematical work.