The given function is:

$f(x) = \frac{3^x}{1 - e^{-4x}}$

1. Determining Continuity:

To determine where the function is continuous, we need to identify where it is defined. This function has a potential issue at points where the denominator becomes zero since division by zero is undefined.

The denominator $1 - e^{-4x}$ is zero when:

$e^{-4x} = 1$ $-4x = 0$ $x = 0$

Thus, $x = 0$ is a point of discontinuity. We also need to check the behavior of $3^x$ and $e^{-4x}$, but since exponential functions are defined for all real numbers, the only issue is at $x = 0$ from the denominator.

Conclusion: $f(x)$ is continuous on $(-\infty, 0) \cup (0, \infty)$.

2. Analyzing the Limits:

Limit as $x$ approaches 0 from the left ($x \to 0^-$):

As $x$ approaches 0 from the left, $e^{-4x}$ becomes very large since $-4x$ becomes a large positive number. This makes $1 - e^{-4x}$ approach a large negative number, and since $3^x$ approaches 1 (as exponential functions are continuous), we have:

$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{3^x}{1 - e^{-4x}} = \frac{1}{-\infty} = 0$

Limit as $x$ approaches 0 from the right ($x \to 0^+$):

As $x$ approaches 0 from the right, $e^{-4x}$ becomes very small since $-4x$ becomes a large negative number. Therefore, $1 - e^{-4x}$ approaches 1, and $3^x$ also approaches 1:

$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{3^x}{1 - e^{-4x}} = \frac{1}{1} = 1$

Summary:

The function is continuous on $(-∞, 0) \cup (0, ∞)$. The left-hand limit as $x$ approaches 0 is 0, and the right-hand limit as $x$ approaches 0 is 1.

Would you like more details on this or any other questions?

Some questions you might consider:

1. How would the graph of this function look near $x = 0$?
2. What is the behavior of the function as $x$ approaches negative or positive infinity?
3. How would the addition of a constant term in the denominator affect the limits?
4. Can we differentiate this function to find its critical points?
5. What is the significance of the base of the exponent (3 and -4) in this function's behavior?

Tip: When analyzing functions involving exponents and their limits, considering their behavior around points of discontinuity can often reveal interesting properties about the function’s overall behavior.