To ensure continuity of the function $f(x)$ at $x = 0$, we need to have:

$\lim_{x \to 0^-} f(x) = f(0) = \lim_{x \to 0^+} f(x)$

1. Function Definitions:

For $f(x)$, the piecewise definition is:

• $f(x) = 5x^2 + a^2 - a$ for $-\frac{\pi}{2} < x < 0$
• $f(0) = b$ for $x = 0$
• $f(x) = 3(1 - \tan x) + 3(1 - \cot x)$ for $0 < x < \frac{\pi}{2}$

2. Left-hand Limit at $x = 0$:

For $-\frac{\pi}{2} < x < 0$, $f(x) = 5x^2 + a^2 - a$.

Taking the limit as $x \to 0^-$ (from the left), we get:

$\lim_{x \to 0^-} f(x) = 5(0)^2 + a^2 - a = a^2 - a$

3. Right-hand Limit at $x = 0$:

For $0 < x < \frac{\pi}{2}$, $f(x) = 3(1 - \tan x) + 3(1 - \cot x)$.

Taking the limit as $x \to 0^+$ (from the right), we get:

$\lim_{x \to 0^+} f(x) = 3(1 - \tan 0) + 3(1 - \cot 0)$

Since $\tan 0 = 0$ and $\cot 0 = \infty$, the second term $3(1 - \cot 0)$ becomes problematic, implying that we need to handle this limit more carefully. However, as $x \to 0$, we can use the small-angle approximations for $\tan x$ and $\cot x$:

$\tan x \approx x, \quad \cot x \approx \frac{1}{x}$

So,

$\lim_{x \to 0^+} f(x) = 3(1 - x) + 3\left(1 - \frac{1}{x}\right)$

This expression goes to infinity because of the $\frac{1}{x}$ term.

Thus, for the function to be continuous at $x = 0$, we must resolve the discontinuity at $x = 0$.

However, since the right-hand limit does not exist in a finite form, it's clear that the function is not continuous unless there's a specific manipulation or additional condition provided (for example, limiting the domain to avoid the infinite behavior).

4. Conclusion:

Given the behavior of $f(x)$ at $x = 0$, continuity is impossible under the current formulation because of the term involving $\cot x$. Would you like to explore an alternative approach to making the function continuous (such as redefining the right-hand part)?

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Here are some related questions you might find helpful:

1. How can we handle infinite terms in limits?
2. What is the significance of the continuity condition for piecewise functions?
3. How do we apply the small-angle approximation to limit problems?
4. What are the properties of the tangent and cotangent functions near zero?
5. How would you modify the function to remove the infinite behavior at $x = 0$?

Tip: In piecewise functions, ensuring continuity often requires that both the left-hand and right-hand limits equal the function’s value at the point. If the limits do not match, the function cannot be continuous at that point.