If `f(x) = {{:((|x+2|)/(tan^(-1)(x+2)), x ne -2),(2, x =-2):}`,

Options

A. continuous at x = - 2
B. not continuous at x = - 2
C. differentiable at x = - 2
D. continuous but not differentiable at x = - 2.

Answer

To solve the problem, we need to determine the continuity of the function defined as:

\[ f(x) = \begin{cases} \frac{|x+2|}{\tan^{-1}(x+2)} & \text{if } x \neq -2 \\ 2 & \text{if } x = -2 \end{cases} \]

### Step 1: Analyze the function at \( x = -2 \)

To check the continuity of \( f(x) \) at \( x = -2 \), we need to evaluate the left-hand limit and the right-hand limit as \( x \) approaches \(-2\).

### Step 2: Calculate the Left-Hand Limit

The left-hand limit as \( x \) approaches \(-2\) is given by:

\[ \lim_{h \to 0^-} f(-2 + h) = \lim_{h \to 0^-} \frac{|(-2 + h) + 2|}{\tan^{-1}((-2 + h) + 2)} \]

This simplifies to:

\[ \lim_{h \to 0^-} \frac{|h|}{\tan^{-1}(h)} \]

Since \( h \) is approaching \( 0 \) from the left, \( |h| = -h \). Thus, we have:

\[ \lim_{h \to 0^-} \frac{-h}{\tan^{-1}(h)} \]

Using the fact that \(\lim_{h \to 0} \frac{h}{\tan^{-1}(h)} = 1\), we get:

\[ \lim_{h \to 0^-} \frac{-h}{\tan^{-1}(h)} = -1 \]

### Step 3: Calculate the Right-Hand Limit

Now, we calculate the right-hand limit as \( x \) approaches \(-2\):

\[ \lim_{h \to 0^+} f(-2 + h) = \lim_{h \to 0^+} \frac{|(-2 + h) + 2|}{\tan^{-1}((-2 + h) + 2)} \]

This simplifies to:

\[ \lim_{h \to 0^+} \frac{h}{\tan^{-1}(h)} \]

Again, using the limit property, we have:

\[ \lim_{h \to 0^+} \frac{h}{\tan^{-1}(h)} = 1 \]

### Step 4: Compare the Limits and the Function Value

Now we compare the left-hand limit, right-hand limit, and the value of the function at \( x = -2 \):

- Left-hand limit: \( -1 \)
- Right-hand limit: \( 1 \)
- \( f(-2) = 2 \)

Since the left-hand limit does not equal the right-hand limit, we conclude that:

\[ \text{Left-hand limit} \neq \text{Right-hand limit} \]

### Conclusion

Since the left-hand limit and right-hand limit are not equal, the function \( f(x) \) is not continuous at \( x = -2 \). Therefore, it is also not differentiable at that point.

### Final Result

The function \( f(x) \) is **not continuous** and **not differentiable** at \( x = -2 \).

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