Question

Use a calculator to graph the function and estimate the value of the limit, then use L'Hôpital's rule to find the limit directly. $$ \lim _{x \rightarrow 0^{+}} \tan \left(x^{x}\right) $$

Short answer

The limit is \( \tan(1) \).

Step-by-step solution

Understand the Problem

We're asked to find the limit \( \lim _{x \rightarrow 0^{+}} \tan(x^{x}) \) using a calculator to graph the function and then applying L'Hôpital's rule for confirmation.

Graph the Function

Use a graphing calculator or software to plot the function \( y = \tan(x^x) \). Evaluate the graph near \( x = 0^+ \). From the graph, notice that as \( x \to 0^+ \), \( x^x \to 1 \), hence \( \tan(x^x) \to \tan(1) \).

Evaluate x^x as x Approaches 0

Remember that \( x^x = e^{x \ln(x)} \). To evaluate \( \lim _{x \rightarrow 0^{+}} x^x \), consider \( \lim _{x \rightarrow 0^{+}} x \ln(x) = \lim _{x \rightarrow 0^{+}} \frac{\ln(x)}{1/x} \).

Apply L'Hôpital's Rule

To find \( \lim _{x \rightarrow 0^{+}} \frac{\ln(x)}{1/x} \), recognize it is an indeterminate form \( \frac{-\infty}{\infty} \). Apply L'Hôpital's Rule: take the derivative of the numerator \( \ln(x) \) and the denominator \( 1/x \), getting \( \frac{1/x}{-1/x^2} = -x \). Then, evaluate \( \lim _{x \rightarrow 0^{+}} -x = 0 \), meaning \( \lim _{x \rightarrow 0^{+}} x \ln(x) = 0 \). Thus, \( \lim _{x \rightarrow 0^{+}} x^x = e^0 = 1 \).

Find the Limit of tan(x^x)

Now that we know \( \lim _{x \rightarrow 0^{+}} x^x = 1 \), the limit \( \lim _{x \rightarrow 0^{+}} \tan(x^x) = \tan(1) \). Evaluate the limit directly to confirm the graphing calculator's estimation.

Final Step: Verify the Result

Revisit the graph and note that \( \tan(1) \approx 1.5574 \) matches our calculations. The graph should converge to this approximate value.

Key concepts

Limits and Continuity

When dealing with calculus, two fundamental concepts are limits and continuity. A *limit* helps identify the value a function approaches as the input gets closer to a certain point. For example, in the given exercise, we're examining how the function \( \tan(x^x) \) behaves as \( x \) approaches 0 from the positive side. We express this with the notation \( \lim_{x \to 0^{+}} \tan(x^x) \). The emphasis on \( 0^{+} \) indicates we're approaching zero from values greater than zero.*Continuity* concerns whether a function can be drawn without any interruptions or holes. If a function is continuous at a point, the limit of the function as \( x \) approaches that point will equal the value of the function at that point. In our scenario, \( \tan(x^x) \) is not straightforwardly continuous at \( x = 0 \) due to the form \( x^x \), but we can evaluate the limit to determine its behavior very close to zero. Understanding both limits and continuity is crucial since they lay the foundation for more advanced topics in calculus, such as differentiability and integral calculus.

Graphing Calculator Usage

Graphing calculators are powerful tools that help visualize mathematical functions. In exercises like the one given, they assist in making educated guesses about the behavior of functions near critical points, thereby supplementing analytical methods.To graph \( y = \tan(x^x) \), you can input it into your calculator by using the substitution \( x^x = e^{x \ln(x)} \). Carefully observing the graph near \( x = 0^{+} \) helps gather insights into the function's limit behavior. You should note important characteristics like slope or curvature.Here are some tips for effectively using a graphing calculator:

• Zoom in: Focus closely on the area of interest to observe small-scale behavior.
• Check endpoints: Ensure you are observing the function as \( x \) approaches from the correct direction.
• Validation: Use graphical insights to confirm findings obtained through algebraic manipulations.

By correlating visual information from graphs with analytic methods, you bolster your understanding of complex mathematical problems.

Indeterminate Forms

Indeterminate forms occur in calculus when limits don't definitively point to a single outcome without additional processing. Common forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), and \( \frac{-\infty}{\infty} \). In this exercise, we encounter it when trying to evaluate \( \lim_{x \to 0^{+}} \frac{\ln(x)}{1/x} \), which at first appears as \( \frac{-\infty}{\infty} \).This is where L'Hôpital's Rule comes in handy. For limits presenting these forms, L'Hôpital's Rule allows us to differentiate the numerator and the denominator separately, subsequently evaluating the limit of the derived fractions. Applying this rule helps simplify these indeterminate forms into more recognizable expressions that can then be easily solved.Here’s how it aids in this problem:

• Take derivatives: Differentiate the numerator and denominator separately.
• Solve the new limit: The new expression resulting from differentiation often simplifies the evaluation process.
• Reanalyze: It provides clarity in expressions that might seem awkward or complex initially.

Recognizing and managing indeterminate forms effectively, especially with the use of tools like L'Hôpital's Rule, is a core skill in mastering calculus concepts.