Problem 17 - Precalculus with Limits

Question

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to 2} \dfrac{x^5-32}{x-2}$$

Step-by-Step Solution

Answer

The limit of the function as x approaches 2 is 80.

Step 1: Identify the indeterminate form

This function results in an indeterminate form when \(x=2\). You can prove it by substituting 2 into the function, giving \(\dfrac{2^5-32}{2-2} = \frac{0}{0}\).

Step 2: Apply L'Hôpital's Rule

Apply L'Hôpital's Rule, which means taking the derivative of the numerator and the denominator. Derivative of the numerator: \(5x^4\), Derivative of the denominator: \(1\). Create the new equation: \(\lim_{x \to 2} \dfrac{5x^4}{1}\).

Step 3: Find the limit

Now find the limit as x approaches 2 in the new equation. This is done by plugging in 2 into the equation, resulting in: \(5*2^{4} = 80\).

Step 4: Verify graphically

Finally, use a graphing utility to plot the function and verify that it appears to approach 80 as x approaches 2.

Step 5: Conclusion

So, the limit of the given function as x approaches 2 is 80.

Key Concepts

L'Hôpital's Rule

When evaluating limits, you might encounter expressions that result in an indeterminate form, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These forms themselves do not provide clear information about the limit. That's where L'Hôpital's Rule becomes handy. L'Hôpital's Rule helps us evaluate limits by allowing us to differentiate both the numerator and the denominator of the problematic expression and then find the limit of this new function instead:

• If you have \(\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{0}{0}\) or \(\frac{\infty}{\infty}\), you can apply L'Hôpital's Rule.
• Take the derivatives of both \(f(x)\) and \(g(x)\) to create a new fraction \(\lim_{{x \to a}} \frac{f'(x)}{g'(x)}\).
• Compute the limit of this new expression, if possible. Keep in mind that you might need to repeat the application of L'Hôpital's Rule if the result is still an indeterminate form.

This rule simplifies the process of finding limits significantly, making it a powerful tool in calculus for handling indeterminate forms.

Indeterminate Forms

Indeterminate forms occur in limits when the initial substitution results in expressions that do not define a clear outcome. The two most common indeterminate forms are \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). When you encounter these, they signal that further analysis is needed to find an accurate limit.Let's explore the key indeterminate forms:

• \(\frac{0}{0}\): This form arises when both the numerator and denominator approach zero. It suggests that a deeper interaction between the functions affects the resulting value.
• \(\frac{\infty}{\infty}\): This happens when both the numerator and the denominator tend towards infinity. It implies that the rates of growth or decrease matter in determining the limit.

There are other less common indeterminate forms, like \(0\cdot\infty\), \(\frac{\infty}{0}\), \(\infty - \infty\), and \(0^0\), which often require algebraic manipulation or special rules for evaluation.Identifying indeterminate forms is crucial because they dictate the method required to resolve the limit, often leading to the use of L'Hôpital's Rule or polynomial elimination to simplify expressions.

Graphical Verification

Graphical verification of limits involves using graphing tools or software to visually confirm the behavior of a function as it approaches a certain point. This step provides a practical check of your analytical work.Here's how graphical verification works:

• Graph the original function \(f(x)\).
• Observe the behavior of the graph near the point of interest. For example, in our exercise, as \(x\) approaches 2.
• Look for the point where the function is heading to verify the limit analytically calculated. If the graph approaches a horizontal line as \(x\) nears the value in question, then your analytical limit likely matches the visual observation.

Graphical verification provides a visual representation that can confirm or contest your calculated limits. It acts as an additional step to ensure accuracy and deepen your understanding of function behaviors near points of interest. It's important to note, however, that graphical methods might not provide the precision or full detail needed for complicated functions, where numerical or algebraic methods are essential.