Problem 83 - Thomas Calculus

Question

Use a CAS to perform the following steps: a. Plot the function near the point \(x_{0}\) being approached. b. From your plot guess the value of the limit. $$\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}$$

Step-by-Step Solution

Answer

The limit is 32.

Step 1: Understand the Problem

We need to find the limit \( \lim_{x \to 2} \frac{x^4 - 16}{x - 2} \). The numerator \( x^4 - 16 \) can be factored since \( 16 \) is a perfect square: \( x^4 - 16 = (x^2 - 4)(x^2 + 4) \). Further factor \( x^2 - 4 \) into \( (x-2)(x+2) \). So, the expression becomes \( \frac{(x-2)(x+2)(x^2+4)}{x-2} \).

Step 2: Simplify the Expression

Cancel out the \( x-2 \) term in the numerator and denominator which gives \( (x+2)(x^2+4) \) for \( x eq 2 \). Now the expression \( (x+2)(x^2+4) \) is easier to evaluate as \( x \to 2 \).

Step 3: Plot the Function

Using a CAS, plot the function \( f(x) = (x+2)(x^2+4) \) near \( x = 2 \). Look for behavior around \( x = 2 \) to understand the function's tendency towards a limit.

Step 4: Guess the Limit from the Plot

Observe the plot closely around \( x = 2 \). The function looks linear near \( x = 2 \) as there are no discontinuities after simplification. Read off the y-value at \( x = 2 \) to make an educated guess of the limit.

Step 5: Calculate Limit with Simplified Expression

Substitute \( x = 2 \) into the simplified expression \((x+2)(x^2+4)\). This gives us \((2+2)(2^2+4) = 4 \times 8 = 32\).

Step 6: Conclusion

Hence, based on both the plot and the computation, the limit can be confirmed to be \( 32 \).

Key Concepts

Factoring Polynomials

Factoring polynomials is one of the essential skills in algebra, especially when dealing with limits involving rational functions. In the exercise, we have the expression \(x^4 - 16\). Notice the similarity to a difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\). Here, \(x^4 - 16\) becomes \((x^2)^2 - 4^2\). This can be rewritten as \((x^2 - 4)(x^2 + 4)\).

• First, recognize \(16\) as a perfect square.
• Apply the formula for the difference of squares.
• Break down further by factoring \(x^2 - 4\) into \((x-2)(x+2)\).

This factoring is crucial to cancel out terms and simplify the limit under study. Remember that factoring can uncover hidden discontinuities and help in simplifying expressions.

Continuity and Discontinuity

Understanding continuity and discontinuity helps in predicting the behavior of functions at certain points. Initially, the expression \(\frac{x^4 - 16}{x-2}\) is undefined at \(x = 2\) since the denominator equals zero. This indicates a potential discontinuity.
However, by factoring and simplifying to \((x+2)(x^2+4)\), we remove the discontinuity at \(x = 2\). This process reveals that the problem was not a vertical asymptote but a removable discontinuity, which means the limit exists. The function becomes continuous once simplified, showing no sudden jumps or breaks.
Always check if factoring a polynomial can resolve discontinuities into a simpler continuous function, making limits easier to evaluate.

Evaluating Limits Using Algebraic Manipulation

Algebraic manipulation is a powerful tool for evaluating limits, especially when functions are indeterminate. The original function \(\frac{x^4 - 16}{x-2}\) initially appears difficult to evaluate at \(x = 2\).
After factoring and simplifying, the expression becomes \((x+2)(x^2+4)\).

• Simplify expressions by factoring out common terms.
• Cancel terms strategically to resolve indeterminate forms.
• Substitute the value you are approaching, here \(x = 2\).

Substituting \(x = 2\) in the simplified expression gives \(4 \times 8 = 32\). This demonstrates the importance of algebraic manipulation in effectively solving limit problems.

Plotting Functions with CAS

CAS tools like graphing calculators or computer software help visualize functions, providing insights into their behavior at different points. In this exercise, plotting the simplified function \(f(x) = (x+2)(x^2+4)\) near \(x = 2\) helps predict the limit visually.

• Use CAS to graph the function accurately around the point of interest.
• Zoom in on the graph to observe the function's behavior right near \(x = 2\).
• Note the y-value that the function approaches as \(x\) nears \(2\).

This practical step solidifies the calculation by showing no breaks or ominous approaches to infinity, thus confirming continuity and consistent behavior of the limit. By observing the smooth and continuous plot near \(x = 2\), the predicted limit of \(32\) makes intuitive sense.