https://file.aiquickdraw.com/mathbot/file/1737452442487u7bu8tik.jpg

The function described in the image appears to be:

\[
f: \mathbb{R} \to \mathbb{R}, \quad f(x) = \frac{\tan(x)}{x}, \quad \text{for} \, x \neq 0
\]
\[
f(0) = 2
\]

To study the continuity of this function, we need to check the behavior at \( x = 0 \), as the function is piecewise defined. For \( x \neq 0 \), the function is given by \( f(x) = \frac{\tan(x)}{x} \), which is continuous for all \( x \neq 0 \).

At \( x = 0 \), we have:

- The function value \( f(0) = 2 \)
- To check continuity at \( x = 0 \), we need to compute the limit of \( f(x) \) as \( x \to 0 \).

Thus, we need to calculate the following limit:

\[
\lim_{x \to 0} \frac{\tan(x)}{x}
\]

Using L'Hopital's Rule or the known standard limit for small \( x \), we have:

\[
\lim_{x \to 0} \frac{\tan(x)}{x} = 1
\]

Therefore, the limit of \( f(x) \) as \( x \to 0 \) is 1, but the function is defined as \( f(0) = 2 \).

Since the limit \( \lim_{x \to 0} f(x) = 1 \) does not equal the function value at \( x = 0 \) (which is 2), the function is **not continuous at \( x = 0 \)**.

If you need further details or clarifications, feel free to ask!

Here are five related questions you might want to explore:

1. What does L'Hopital's Rule state, and how is it applied to limits like the one above?
2. How can you determine the continuity of a piecewise function?
3. What is the definition of continuity at a point?
4. What other standard limits involving trigonometric functions are useful in calculus?
5. Can you find the derivative of \( f(x) \) at \( x = 0 \)?

**Tip:** Always check both the function value at a point and the limit approaching that point when testing continuity.

Explore the continuity of the piecewise function f(x) = tan(x)/x for x ≠ 0 and f(0) = 2. This problem involves key mathematical concepts such as continuity, L'Hopital's Rule, and piecewise functions. Suitable for Grades 11-12.

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