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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.

Use continuity to evaluate lim (x -> 1) e^(x^2 - 3x - 3)

Learn how to evaluate the limit of the function e^(x^2 - 3x - 3) as x approaches 1 using continuity. This solution covers key concepts in limits, continuity, and exponential functions. Ideal for students in Grades 10-12.

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