To evaluate the limit $\lim_{x \to 1} e^{x^2 - 3x - 3}$ using continuity, we follow these steps:

1. Recognize the function is continuous: The exponential function $e^x$ is continuous everywhere, so we can directly substitute the value $x = 1$ into the exponent.

2. Substitute $x = 1$ into the exponent: $x^2 - 3x - 3 = (1)^2 - 3(1) - 3 = 1 - 3 - 3 = -5$

3. Evaluate the exponential: $e^{x^2 - 3x - 3} = e^{-5}$

Thus, the value of the limit is: $\lim_{x \to 1} e^{x^2 - 3x - 3} = e^{-5}$

Would you like more details on this process or further explanation?

Here are 5 related questions for further practice:

1. What is the limit of $\lim_{x \to 2} e^{x^2 - 3x - 3}$?
2. How do we apply the property of continuity to evaluate limits of other transcendental functions?
3. Can we evaluate $\lim_{x \to 1} \ln(x^2 - 3x - 3)$?
4. What is the derivative of the function $f(x) = e^{x^2 - 3x - 3}$?
5. How does the limit change if the exponent was $x^3 - 3x^2 - 3$?

Tip: Always remember that if a function is continuous at a point, you can directly substitute that point into the function to evaluate the limit.