graph the function over -2pi less than or equal to x 2pi greater than or equal to x using technology. use the graph to estimate the value of lim(gx) as x approaches 0. then justify answer using algebra

(4x+x^3sin(2pix+pi))/(6x)

The graph of the function \( g(x) = \frac{4x + x^3 \sin(2\pi x + \pi)}{6x} \) over the interval \([-2\pi, 2\pi]\) is shown above. 

### Step 2: Estimate the Limit from the Graph
From the graph, as \( x \) approaches 0, the value of \( g(x) \) appears to approach \(\frac{2}{3}\).

### Step 3: Justify the Limit Algebraically
To find the limit algebraically, we calculate:

\[
\lim_{x \to 0} g(x) = \lim_{x \to 0} \frac{4x + x^3 \sin(2\pi x + \pi)}{6x}
\]

We can separate the limit into two parts:

\[
\lim_{x \to 0} g(x) = \lim_{x \to 0} \left(\frac{4x}{6x} + \frac{x^3 \sin(2\pi x + \pi)}{6x}\right)
\]

Simplifying each term:

\[
\lim_{x \to 0} \frac{4x}{6x} = \frac{4}{6} = \frac{2}{3}
\]

For the second term:

\[
\lim_{x \to 0} \frac{x^3 \sin(2\pi x + \pi)}{6x} = \lim_{x \to 0} \frac{x^2 \sin(2\pi x + \pi)}{6}
\]

As \( x \to 0 \), \( \sin(2\pi x + \pi) \) is bounded, so \( x^2 \sin(2\pi x + \pi) \) approaches 0. Therefore:

\[
\lim_{x \to 0} \frac{x^2 \sin(2\pi x + \pi)}{6} = 0
\]

Thus, the overall limit is:

\[
\lim_{x \to 0} g(x) = \frac{2}{3} + 0 = \frac{2}{3}
\]

This confirms the estimate from the graph.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do limits behave when dealing with trigonometric functions?
2. What is L'Hôpital's Rule, and when can it be used for limits?
3. How do you determine the continuity of a function at a point?
4. What role does the squeeze theorem play in evaluating limits?
5. How can you verify a limit algebraically when the function involves polynomials and trigonometric terms?

**Tip:** When estimating limits from a graph, always check for continuity and smooth behavior around the point of interest to ensure accuracy.

Explore how to estimate and justify the limit of a complex trigonometric function, \( g(x) = \frac{4x + x^3 \sin(2\pi x + \pi)}{6x} \), as x approaches 0. Includes step-by-step graph analysis and algebraic justification. Suitable for advanced high school students and learners of calculus.

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